Algorithmic Graph Theory (SS2016) Chapter 1 Planar Graphs Walter Unger
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(2) Basic Definitions 1. Introduction to planar Graphs. Separators. Inhaltsverzeichnis. Contents I. 1. Basic Definitions Graphs Special Graphs Connectivity of Graphs Statements. 2. Introduction to planar Graphs Definitions Theorems on planar Graphs Definitions on outer-planar graphs Theorems on outer-planar Graphs Theorems on SP-Graphs Homeomorph Graphs. Z. Applications Walter Unger 21.12.2018 12:22. 3. Separators Motivation Definition Examples Alternative Definition Introduction to planar Separators Overview Preparation Overview Planare-Graph-Separator Theorem. 4. Applications Independent Set on planar Graphs. SS2016.
(3) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 1/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(4) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 2/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(5) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 3/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(6) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 4/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(7) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 5/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(8) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 6/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(9) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 7/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(10) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 8/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(11) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 9/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(12) Basic Definitions 1:1. Graphs. Introduction to planar Graphs. Separators. 10/10. Definition: Graph v2. v9. Definition (Undirected Graph) Let V (G ) = {v1 , ..., vn } be a non-empty set of nodes and E (G ) be a set or multiset of pairs from V (G ) (set of edges). The sets V (G ) and E (G ) define the graph G = (V (G ), E (G )). If G is uniquely determined, then we just write: V and E . Or in other words G = (V , E ). We always use as default writing: n = |V | and m = |E |.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(13) Basic Definitions 1:2. Graphs. Introduction to planar Graphs. Separators. 1/9. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(14) Basic Definitions 1:2. Graphs. Introduction to planar Graphs. Separators. 2/9. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(15) Basic Definitions 1:2. Graphs. Introduction to planar Graphs. Separators. 3/9. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(16) Basic Definitions 1:2. Graphs. Introduction to planar Graphs. Separators. 4/9. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(17) Basic Definitions 1:2. Graphs. Introduction to planar Graphs. Separators. 5/9. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(18) Basic Definitions 1:2. Graphs. Introduction to planar Graphs. Separators. 6/9. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(19) Basic Definitions 1:2. Graphs. Introduction to planar Graphs. Separators. 7/9. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(20) Basic Definitions 1:2. Graphs. Introduction to planar Graphs. Separators. 8/9. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(21) Basic Definitions 1:2. Graphs. Introduction to planar Graphs. Separators. 9/9. Way of Speaking for Graphs v2. v9. Definition (Way of Speaking) Let G = (V (G ), E (G )) and e = (v , w ) ∈ E (G ). The nodes v , w are called connected (adjacent) by an edge e.. v4 v6 v1 v8. An edge e is called loop, if v = w holds.. v5. Two edges are called parallel, if they are the same. A graph without parallel edges is called simple.. v0 v3. As long as we do not state differently we will use in the following simple graph without loops.. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(22) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 1/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(23) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 2/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(24) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 3/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(25) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 4/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(26) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 5/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(27) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 6/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(28) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 7/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(29) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 8/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(30) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 9/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(31) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 10/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(32) Basic Definitions 1:3. Graphs. Introduction to planar Graphs. Separators. 11/11. Degree of a Node Definition (Degree of a Node) Let v ∈ V (G ). With deg(v ) = |{e ∈ E (G ) e = (v , v 0 ), v 0 ∈ V (G ) \ {v }}| we denote the degree of a Node (degree) of v . v2. deg(v0 ) = 4.. v9. deg(v1 ) = 3.. v4. deg(v4 ) = 6. deg(v5 ) = 6.. v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(33) Basic Definitions 1:4. Special Graphs. Introduction to planar Graphs. Separators. 1/4. Regular and Complete Definition (Regular) A graph G is called k-regular, iff for all v ∈ V (G ) we have: d(v ) = k.. e d. f. a. c b. Definition (Complete) A graph G is called complete, iff all pairs of nodes a, b from V holds: (a, b) ∈ E . Notation: Kn .. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(34) Basic Definitions 1:4. Special Graphs. Introduction to planar Graphs. Separators. 2/4. Regular and Complete Definition (Regular) A graph G is called k-regular, iff for all v ∈ V (G ) we have: d(v ) = k.. e d. a b. e f. d. c. a. f. c b. Definition (Complete) A graph G is called complete, iff all pairs of nodes a, b from V holds: (a, b) ∈ E . Notation: Kn .. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(35) Basic Definitions 1:4. Special Graphs. Introduction to planar Graphs. Separators. 3/4. Regular and Complete Definition (Regular) A graph G is called k-regular, iff for all v ∈ V (G ) we have: d(v ) = k.. e d. a b. e f. d. c. a b. e f. d. c. a. f. c b. Definition (Complete) A graph G is called complete, iff all pairs of nodes a, b from V holds: (a, b) ∈ E . Notation: Kn .. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(36) Basic Definitions 1:4. Special Graphs. Introduction to planar Graphs. Separators. 4/4. Regular and Complete Definition (Regular) A graph G is called k-regular, iff for all v ∈ V (G ) we have: d(v ) = k.. e d. a b. e f. d. c. a b. e f. d. c. a. f. c b. Definition (Complete) A graph G is called complete, iff all pairs of nodes a, b from V holds: (a, b) ∈ E . Notation: Kn .. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(37) Basic Definitions 1:5. Special Graphs. Introduction to planar Graphs 1/7. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Special Graphs. SS2016. Definition (Bipartite) A Graph G is called bipartite, iff V may be split in to disjoint set V 0 , V 00 , such that each edge connects only nodes from both partitions. Notation: G = (V 0 , V 00 , E ) Definition (Complete bipartite) A Graph G is called complete bipartite, iff V may be split in to disjoint set V 0 , V 00 , and E = {(a, b) a ∈ V 0 , b ∈ V 00 }. Notation: Kp,q with p = |V 0 | and q = |V 00 |. Star, iff Sn = K1,n−1 ..
(38) Basic Definitions 1:5. Special Graphs. Introduction to planar Graphs 2/7. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Special Graphs. SS2016. Definition (Bipartite) A Graph G is called bipartite, iff V may be split in to disjoint set V 0 , V 00 , such that each edge connects only nodes from both partitions. Notation: G = (V 0 , V 00 , E ) Definition (Complete bipartite) A Graph G is called complete bipartite, iff V may be split in to disjoint set V 0 , V 00 , and E = {(a, b) a ∈ V 0 , b ∈ V 00 }. Notation: Kp,q with p = |V 0 | and q = |V 00 |. Star, iff Sn = K1,n−1 ..
(39) Basic Definitions 1:5. Special Graphs. Introduction to planar Graphs 3/7. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Special Graphs. SS2016. Definition (Bipartite) A Graph G is called bipartite, iff V may be split in to disjoint set V 0 , V 00 , such that each edge connects only nodes from both partitions. Notation: G = (V 0 , V 00 , E ) Definition (Complete bipartite) A Graph G is called complete bipartite, iff V may be split in to disjoint set V 0 , V 00 , and E = {(a, b) a ∈ V 0 , b ∈ V 00 }. Notation: Kp,q with p = |V 0 | and q = |V 00 |. Star, iff Sn = K1,n−1 ..
(40) Basic Definitions 1:5. Special Graphs. Introduction to planar Graphs 4/7. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Special Graphs. SS2016. Definition (Bipartite) A Graph G is called bipartite, iff V may be split in to disjoint set V 0 , V 00 , such that each edge connects only nodes from both partitions. Notation: G = (V 0 , V 00 , E ) Definition (Complete bipartite) A Graph G is called complete bipartite, iff V may be split in to disjoint set V 0 , V 00 , and E = {(a, b) a ∈ V 0 , b ∈ V 00 }. Notation: Kp,q with p = |V 0 | and q = |V 00 |. Star, iff Sn = K1,n−1 ..
(41) Basic Definitions 1:5. Special Graphs. Introduction to planar Graphs 5/7. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Special Graphs. SS2016. Definition (Bipartite) A Graph G is called bipartite, iff V may be split in to disjoint set V 0 , V 00 , such that each edge connects only nodes from both partitions. Notation: G = (V 0 , V 00 , E ) Definition (Complete bipartite) A Graph G is called complete bipartite, iff V may be split in to disjoint set V 0 , V 00 , and E = {(a, b) a ∈ V 0 , b ∈ V 00 }. Notation: Kp,q with p = |V 0 | and q = |V 00 |. Star, iff Sn = K1,n−1 ..
(42) Basic Definitions 1:5. Special Graphs. Introduction to planar Graphs 6/7. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Special Graphs. SS2016. Definition (Bipartite) A Graph G is called bipartite, iff V may be split in to disjoint set V 0 , V 00 , such that each edge connects only nodes from both partitions. Notation: G = (V 0 , V 00 , E ) Definition (Complete bipartite) A Graph G is called complete bipartite, iff V may be split in to disjoint set V 0 , V 00 , and E = {(a, b) a ∈ V 0 , b ∈ V 00 }. Notation: Kp,q with p = |V 0 | and q = |V 00 |. Star, iff Sn = K1,n−1 ..
(43) Basic Definitions 1:5. Special Graphs. Introduction to planar Graphs 7/7. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Special Graphs. SS2016. Definition (Bipartite) A Graph G is called bipartite, iff V may be split in to disjoint set V 0 , V 00 , such that each edge connects only nodes from both partitions. Notation: G = (V 0 , V 00 , E ) Definition (Complete bipartite) A Graph G is called complete bipartite, iff V may be split in to disjoint set V 0 , V 00 , and E = {(a, b) a ∈ V 0 , b ∈ V 00 }. Notation: Kp,q with p = |V 0 | and q = |V 00 |. Star, iff Sn = K1,n−1 ..
(44) Basic Definitions 1:6. Special Graphs. Introduction to planar Graphs 1/4. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Examples. a4. b4. b4. a3. b3. b3. a2. b2. b2. a1. b1. b1. a0. b0. a0. b0. SS2016.
(45) Basic Definitions 1:6. Special Graphs. Introduction to planar Graphs 2/4. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Examples. a4. b4. b4. a3. b3. b3. a2. b2. b2. a1. b1. a1. b1. a0. b0. a0. b0. SS2016.
(46) Basic Definitions 1:6. Special Graphs. Introduction to planar Graphs 3/4. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Examples. a4. b4. b4. a3. b3. b3. a2. b2. a2. b2. a1. b1. a1. b1. a0. b0. a0. b0. SS2016.
(47) Basic Definitions 1:6. Special Graphs. Introduction to planar Graphs 4/4. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Examples. a4. b4. a3. b3. a3. b3. a2. b2. a2. b2. a1. b1. a1. b1. a0. b0. a0. b0. b4. SS2016.
(48) Basic Definitions 1:7. Special Graphs. Introduction to planar Graphs. Separators. 1/6. Subgraphs v2. v9. Definition (Subgraph) A Graph H = (V (H), E (H)) is call a subgraph of G = (V (G ), E (G )),. v4 v6. iff V (H) ⊆ V (G ) and E (H) ⊆ E (G ). v1. v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(49) Basic Definitions 1:7. Special Graphs. Introduction to planar Graphs. Separators. 2/6. Subgraphs v2. v9. Definition (Subgraph) A Graph H = (V (H), E (H)) is call a subgraph of G = (V (G ), E (G )),. v4 v6. iff V (H) ⊆ V (G ) and E (H) ⊆ E (G ). v1. v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(50) Basic Definitions 1:7. Special Graphs. Introduction to planar Graphs. Separators. 3/6. Subgraphs v2. v9. Definition (Subgraph) A Graph H = (V (H), E (H)) is call a subgraph of G = (V (G ), E (G )),. v4 v6. iff V (H) ⊆ V (G ) and E (H) ⊆ E (G ). v1. v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(51) Basic Definitions 1:7. Special Graphs. Introduction to planar Graphs. Separators. 4/6. Subgraphs v2. v9. Definition (Subgraph) A Graph H = (V (H), E (H)) is call a subgraph of G = (V (G ), E (G )),. v4 v6. iff V (H) ⊆ V (G ) and E (H) ⊆ E (G ). v1. v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(52) Basic Definitions 1:7. Special Graphs. Introduction to planar Graphs. Separators. 5/6. Subgraphs v2. v9. Definition (Subgraph) A Graph H = (V (H), E (H)) is call a subgraph of G = (V (G ), E (G )),. v4 v6. iff V (H) ⊆ V (G ) and E (H) ⊆ E (G ). v1. v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(53) Basic Definitions 1:7. Special Graphs. Introduction to planar Graphs. Separators. 6/6. Subgraphs v2. v9. Definition (Subgraph) A Graph H = (V (H), E (H)) is call a subgraph of G = (V (G ), E (G )),. v4 v6. iff V (H) ⊆ V (G ) and E (H) ⊆ E (G ). v1. v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(54) Basic Definitions 1:8. Special Graphs. Introduction to planar Graphs. Separators. 1/6. Subgraphs v2. v9. Definition (node-induced subgraph) A graph H = (V (H), E (H)) is a node-induced subgraph of G = (V (G ), E (G )), iff V (H) ⊆ V (G ) and E (H) = {(a, b) ∈ E (G ) a, b ∈ V (H)}.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(55) Basic Definitions 1:8. Special Graphs. Introduction to planar Graphs. Separators. 2/6. Subgraphs v2. v9. Definition (node-induced subgraph) A graph H = (V (H), E (H)) is a node-induced subgraph of G = (V (G ), E (G )), iff V (H) ⊆ V (G ) and E (H) = {(a, b) ∈ E (G ) a, b ∈ V (H)}.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(56) Basic Definitions 1:8. Special Graphs. Introduction to planar Graphs. Separators. 3/6. Subgraphs v2. v9. Definition (node-induced subgraph) A graph H = (V (H), E (H)) is a node-induced subgraph of G = (V (G ), E (G )), iff V (H) ⊆ V (G ) and E (H) = {(a, b) ∈ E (G ) a, b ∈ V (H)}.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(57) Basic Definitions 1:8. Special Graphs. Introduction to planar Graphs. Separators. 4/6. Subgraphs v2. v9. Definition (node-induced subgraph) A graph H = (V (H), E (H)) is a node-induced subgraph of G = (V (G ), E (G )), iff V (H) ⊆ V (G ) and E (H) = {(a, b) ∈ E (G ) a, b ∈ V (H)}.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(58) Basic Definitions 1:8. Special Graphs. Introduction to planar Graphs. Separators. 5/6. Subgraphs v2. v9. Definition (node-induced subgraph) A graph H = (V (H), E (H)) is a node-induced subgraph of G = (V (G ), E (G )), iff V (H) ⊆ V (G ) and E (H) = {(a, b) ∈ E (G ) a, b ∈ V (H)}.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(59) Basic Definitions 1:8. Special Graphs. Introduction to planar Graphs. Separators. 6/6. Subgraphs v2. v9. Definition (node-induced subgraph) A graph H = (V (H), E (H)) is a node-induced subgraph of G = (V (G ), E (G )), iff V (H) ⊆ V (G ) and E (H) = {(a, b) ∈ E (G ) a, b ∈ V (H)}.. v4 v6 v1 v8 v5. v0 v3. Z. Applications Walter Unger 21.12.2018 12:22. v7. SS2016.
(60) Basic Definitions 1:9. Connectivity of Graphs. Introduction to planar Graphs. Separators. 1/6. Connectivity. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Definition A graph G = (V , E ) is called connected, iff between any two different nodes a, b exists a path from a to b. v2. v9 v4 v6. v1 v8 v5 v0 v3. v7.
(61) Basic Definitions 1:9. Connectivity of Graphs. Introduction to planar Graphs. Separators. 2/6. Connectivity. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Definition A graph G = (V , E ) is called connected, iff between any two different nodes a, b exists a path from a to b. v2. v9 v4 v6. v1 v8 v5 v0 v3. v7.
(62) Basic Definitions 1:9. Connectivity of Graphs. Introduction to planar Graphs. Separators. 3/6. Connectivity. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Definition A graph G = (V , E ) is called connected, iff between any two different nodes a, b exists a path from a to b. v2. v9 v4 v6. v1 v8 v5 v0 v3. v7.
(63) Basic Definitions 1:9. Connectivity of Graphs. Introduction to planar Graphs. Separators. 4/6. Connectivity. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Definition A graph G = (V , E ) is called connected, iff between any two different nodes a, b exists a path from a to b. v2. v9 v4 v6. v1 v8 v5 v0 v3. v7.
(64) Basic Definitions 1:9. Connectivity of Graphs. Introduction to planar Graphs. Separators. 5/6. Connectivity. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Definition A graph G = (V , E ) is called connected, iff between any two different nodes a, b exists a path from a to b. v2. v9 v4 v6. v1 v8 v5 v0 v3. v7.
(65) Basic Definitions 1:9. Connectivity of Graphs. Introduction to planar Graphs. Separators. 6/6. Connectivity. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Definition A graph G = (V , E ) is called connected, iff between any two different nodes a, b exists a path from a to b. v2. v9 v4 v6. v1 v8 v5 v0 v3. v7.
(66) Basic Definitions 1:10. Connectivity of Graphs. Introduction to planar Graphs 1/5. Separators. Node-Separator Definition Let G = (V , E ), V 0 ⊂ V is called a node-separator (vertex cut), iff G − V 0 is not connected. Notation: G − V 0 := (V \ V 0 , {(a, b) ∈ E | a, b ∈ V \ V 0 }) Definition If {v } is a node-separator, then v is called articulation point. Theorem Only cliques Kn do not have any node-separator.. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(67) Basic Definitions 1:10. Connectivity of Graphs. Introduction to planar Graphs 2/5. Separators. Node-Separator Definition Let G = (V , E ), V 0 ⊂ V is called a node-separator (vertex cut), iff G − V 0 is not connected. Notation: G − V 0 := (V \ V 0 , {(a, b) ∈ E | a, b ∈ V \ V 0 }) Definition If {v } is a node-separator, then v is called articulation point. Theorem Only cliques Kn do not have any node-separator.. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(68) Basic Definitions 1:10. Connectivity of Graphs. Introduction to planar Graphs 3/5. Separators. Node-Separator Definition Let G = (V , E ), V 0 ⊂ V is called a node-separator (vertex cut), iff G − V 0 is not connected. Notation: G − V 0 := (V \ V 0 , {(a, b) ∈ E | a, b ∈ V \ V 0 }) Definition If {v } is a node-separator, then v is called articulation point. Theorem Only cliques Kn do not have any node-separator.. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(69) Basic Definitions 1:10. Connectivity of Graphs. Introduction to planar Graphs 4/5. Separators. Node-Separator Definition Let G = (V , E ), V 0 ⊂ V is called a node-separator (vertex cut), iff G − V 0 is not connected. Notation: G − V 0 := (V \ V 0 , {(a, b) ∈ E | a, b ∈ V \ V 0 }) Definition If {v } is a node-separator, then v is called articulation point. Theorem Only cliques Kn do not have any node-separator.. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(70) Basic Definitions 1:10. Connectivity of Graphs. Introduction to planar Graphs 5/5. Separators. Node-Separator Definition Let G = (V , E ), V 0 ⊂ V is called a node-separator (vertex cut), iff G − V 0 is not connected. Notation: G − V 0 := (V \ V 0 , {(a, b) ∈ E | a, b ∈ V \ V 0 }) Definition If {v } is a node-separator, then v is called articulation point. Theorem Only cliques Kn do not have any node-separator.. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(71) Basic Definitions 1:11. Introduction to planar Graphs. Connectivity of Graphs. Separators Walter Unger 21.12.2018 12:22. Example. v2. v9. v2. v9 v4. v4 v6. v6. v1. v1 v8. v8. v5. v5. v0. v0 v3. v7. v2. v3. v7. v2. v9. v9 v4. v4. v6. v6 v1. v1. v8. v8 v5. v5 v0. v0 v3. v7. Z. Applications. 1/5. v3. v7. SS2016.
(72) Basic Definitions 1:11. Introduction to planar Graphs. Connectivity of Graphs. Separators Walter Unger 21.12.2018 12:22. Example. v2. v9. v2. v9 v4. v4 v6. v6. v1. v1 v8. v8. v5. v5. v0. v0 v3. v7. v2. v3. v7. v2. v9. v9 v4. v4. v6. v6 v1. v1. v8. v8 v5. v5 v0. v0 v3. v7. Z. Applications. 2/5. v3. v7. SS2016.
(73) Basic Definitions 1:11. Introduction to planar Graphs. Connectivity of Graphs. Separators Walter Unger 21.12.2018 12:22. Example. v2. v9. v2. v9 v4. v4 v6. v6. v1. v1 v8. v8. v5. v5. v0. v0 v3. v7. v2. v3. v7. v2. v9. v9 v4. v4. v6. v6 v1. v1. v8. v8 v5. v5 v0. v0 v3. v7. Z. Applications. 3/5. v3. v7. SS2016.
(74) Basic Definitions 1:11. Introduction to planar Graphs. Connectivity of Graphs. Separators Walter Unger 21.12.2018 12:22. Example. v2. v9. v2. v9 v4. v4 v6. v6. v1. v1 v8. v8. v5. v5. v0. v0 v3. v7. v2. v3. v7. v2. v9. v9 v4. v4. v6. v6 v1. v1. v8. v8 v5. v5 v0. v0 v3. v7. Z. Applications. 4/5. v3. v7. SS2016.
(75) Basic Definitions 1:11. Introduction to planar Graphs. Connectivity of Graphs. Separators Walter Unger 21.12.2018 12:22. Example. v2. v9. v2. v9 v4. v4 v6. v6. v1. v1 v8. v8. v5. v5. v0. v0 v3. v7. v2. v3. v7. v2. v9. v9 v4. v4. v6. v6 v1. v1. v8. v8 v5. v5 v0. v0 v3. v7. Z. Applications. 5/5. v3. v7. SS2016.
(76) Basic Definitions 1:12. Connectivity of Graphs. Introduction to planar Graphs 1/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Edge-Separator Definition Let G = (V , E ). E 0 ⊂ E is called edge-separator (edge cut), iff G − E 0 is not connected. Notation: G − E 0 := (V , E \ E 0 ) Definition If {v , w } is an edge-separator, then {v , w } is called a bridge. Theorem An minimal edge-separator E 0 of G = (V , E ) induces a 2-partite graph. Or in other words: G = (V , E 0 ) is a 2-partite graph.. SS2016.
(77) Basic Definitions 1:12. Connectivity of Graphs. Introduction to planar Graphs 2/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Edge-Separator Definition Let G = (V , E ). E 0 ⊂ E is called edge-separator (edge cut), iff G − E 0 is not connected. Notation: G − E 0 := (V , E \ E 0 ) Definition If {v , w } is an edge-separator, then {v , w } is called a bridge. Theorem An minimal edge-separator E 0 of G = (V , E ) induces a 2-partite graph. Or in other words: G = (V , E 0 ) is a 2-partite graph.. SS2016.
(78) Basic Definitions 1:12. Connectivity of Graphs. Introduction to planar Graphs 3/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Edge-Separator Definition Let G = (V , E ). E 0 ⊂ E is called edge-separator (edge cut), iff G − E 0 is not connected. Notation: G − E 0 := (V , E \ E 0 ) Definition If {v , w } is an edge-separator, then {v , w } is called a bridge. Theorem An minimal edge-separator E 0 of G = (V , E ) induces a 2-partite graph. Or in other words: G = (V , E 0 ) is a 2-partite graph.. SS2016.
(79) Basic Definitions 1:12. Connectivity of Graphs. Introduction to planar Graphs 4/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Edge-Separator Definition Let G = (V , E ). E 0 ⊂ E is called edge-separator (edge cut), iff G − E 0 is not connected. Notation: G − E 0 := (V , E \ E 0 ) Definition If {v , w } is an edge-separator, then {v , w } is called a bridge. Theorem An minimal edge-separator E 0 of G = (V , E ) induces a 2-partite graph. Or in other words: G = (V , E 0 ) is a 2-partite graph.. SS2016.
(80) Basic Definitions 1:12. Connectivity of Graphs. Introduction to planar Graphs 5/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Edge-Separator Definition Let G = (V , E ). E 0 ⊂ E is called edge-separator (edge cut), iff G − E 0 is not connected. Notation: G − E 0 := (V , E \ E 0 ) Definition If {v , w } is an edge-separator, then {v , w } is called a bridge. Theorem An minimal edge-separator E 0 of G = (V , E ) induces a 2-partite graph. Or in other words: G = (V , E 0 ) is a 2-partite graph.. SS2016.
(81) Basic Definitions 1:13. Connectivity of Graphs. Introduction to planar Graphs. Separators. 1/2. Example v2. v9. v4 v6 v1 v8 v5. v0 v3. v7. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(82) Basic Definitions 1:13. Connectivity of Graphs. Introduction to planar Graphs. Separators. 2/2. Example v2. v9. v4 v6 v1 v8 v5. v0 v3. v7. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(83) Basic Definitions 1:14. Connectivity of Graphs. Introduction to planar Graphs 1/3. Separators. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Connectivity Definition A Graph G = (V , E ) is called k-connected, iff ∀V 0 ⊂ V : |V 0 | = k − 1 we have G − V 0 is connected. A k-connected Graph is also k − 1-connected. Notation: κ(G ) = k Definition. Let G = (V , E ) and k minimal with: ∃E 0 ⊂ E : |E 0 | = k and G − E 0 is not connected or trivial. Then we call G k-edge-connected. A k-edge-connected Graph is also k − 1-edge-connected. Notation: λ(G ) = k.
(84) Basic Definitions 1:14. Connectivity of Graphs. Introduction to planar Graphs 2/3. Separators. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Connectivity Definition A Graph G = (V , E ) is called k-connected, iff ∀V 0 ⊂ V : |V 0 | = k − 1 we have G − V 0 is connected. A k-connected Graph is also k − 1-connected. Notation: κ(G ) = k Definition. Let G = (V , E ) and k minimal with: ∃E 0 ⊂ E : |E 0 | = k and G − E 0 is not connected or trivial. Then we call G k-edge-connected. A k-edge-connected Graph is also k − 1-edge-connected. Notation: λ(G ) = k.
(85) Basic Definitions 1:14. Connectivity of Graphs. Introduction to planar Graphs 3/3. Separators. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Connectivity Definition A Graph G = (V , E ) is called k-connected, iff ∀V 0 ⊂ V : |V 0 | = k − 1 we have G − V 0 is connected. A k-connected Graph is also k − 1-connected. Notation: κ(G ) = k Definition. Let G = (V , E ) and k minimal with: ∃E 0 ⊂ E : |E 0 | = k and G − E 0 is not connected or trivial. Then we call G k-edge-connected. A k-edge-connected Graph is also k − 1-edge-connected. Notation: λ(G ) = k.
(86) Basic Definitions 1:15. Statements. Introduction to planar Graphs. Separators. 1/5. Statements on Connectivity Theorem For any graph G = (V , E ) we have: κ(G ) 6 λ(G ) 6 δ(G ) Notation: δ(G ) := min{deg(v ) | v ∈ V } Theorem For all integer numbers 0 < a 6 b 6 c there are graphs G with: κ(G ) = a, λ(G ) = b, δ(G ) = c Theorem Let G = (V , E ) be a graph with: |V | = n and δ(G ) > n/2. Then we have: λ(G ) = δ(G ). Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(87) Basic Definitions 1:15. Statements. Introduction to planar Graphs. Separators. 2/5. Statements on Connectivity Theorem For any graph G = (V , E ) we have: κ(G ) 6 λ(G ) 6 δ(G ) Notation: δ(G ) := min{deg(v ) | v ∈ V } Theorem For all integer numbers 0 < a 6 b 6 c there are graphs G with: κ(G ) = a, λ(G ) = b, δ(G ) = c Theorem Let G = (V , E ) be a graph with: |V | = n and δ(G ) > n/2. Then we have: λ(G ) = δ(G ). Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(88) Basic Definitions 1:15. Statements. Introduction to planar Graphs. Separators. 3/5. Statements on Connectivity Theorem For any graph G = (V , E ) we have: κ(G ) 6 λ(G ) 6 δ(G ) Notation: δ(G ) := min{deg(v ) | v ∈ V } Theorem For all integer numbers 0 < a 6 b 6 c there are graphs G with: κ(G ) = a, λ(G ) = b, δ(G ) = c Theorem Let G = (V , E ) be a graph with: |V | = n and δ(G ) > n/2. Then we have: λ(G ) = δ(G ). Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(89) Basic Definitions 1:15. Statements. Introduction to planar Graphs. Separators. 4/5. Statements on Connectivity Theorem For any graph G = (V , E ) we have: κ(G ) 6 λ(G ) 6 δ(G ) Notation: δ(G ) := min{deg(v ) | v ∈ V } Theorem For all integer numbers 0 < a 6 b 6 c there are graphs G with: κ(G ) = a, λ(G ) = b, δ(G ) = c Theorem Let G = (V , E ) be a graph with: |V | = n and δ(G ) > n/2. Then we have: λ(G ) = δ(G ). Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(90) Basic Definitions 1:15. Statements. Introduction to planar Graphs. Separators. 5/5. Statements on Connectivity Theorem For any graph G = (V , E ) we have: κ(G ) 6 λ(G ) 6 δ(G ) Notation: δ(G ) := min{deg(v ) | v ∈ V } Theorem For all integer numbers 0 < a 6 b 6 c there are graphs G with: κ(G ) = a, λ(G ) = b, δ(G ) = c Theorem Let G = (V , E ) be a graph with: |V | = n and δ(G ) > n/2. Then we have: λ(G ) = δ(G ). Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(91) Basic Definitions 1:16. Statements. Introduction to planar Graphs. Separators. 1/6. SS2016. Statements on Node-Connectivity Theorem Let G = (V , E ) with: |V | = n and |E | = m. Then is the maximal connectivity (maximal k with G is k-connected) of G : 0 falls m < n − 1 2 · m/n if m >n−1 Theorem Let G = (V , E ) connected. The following statements are equivalent: 1. v ∈ V is a node-separator.. 2. ∃a, b ∈ V : a, b 6= v : each path from a to b traverses via v . ˙ = V \ {v } and each path from a ∈ A to b ∈ B traverses via v . ∃A, B: A∪B. 3. Z. Applications Walter Unger 21.12.2018 12:22.
(92) Basic Definitions 1:16. Statements. Introduction to planar Graphs. Separators. 2/6. SS2016. Statements on Node-Connectivity Theorem Let G = (V , E ) with: |V | = n and |E | = m. Then is the maximal connectivity (maximal k with G is k-connected) of G : 0 falls m < n − 1 2 · m/n if m >n−1 Theorem Let G = (V , E ) connected. The following statements are equivalent: 1. v ∈ V is a node-separator.. 2. ∃a, b ∈ V : a, b 6= v : each path from a to b traverses via v . ˙ = V \ {v } and each path from a ∈ A to b ∈ B traverses via v . ∃A, B: A∪B. 3. Z. Applications Walter Unger 21.12.2018 12:22.
(93) Basic Definitions 1:16. Statements. Introduction to planar Graphs. Separators. 3/6. SS2016. Statements on Node-Connectivity Theorem Let G = (V , E ) with: |V | = n and |E | = m. Then is the maximal connectivity (maximal k with G is k-connected) of G : 0 falls m < n − 1 2 · m/n if m >n−1 Theorem Let G = (V , E ) connected. The following statements are equivalent: 1. v ∈ V is a node-separator.. 2. ∃a, b ∈ V : a, b 6= v : each path from a to b traverses via v . ˙ = V \ {v } and each path from a ∈ A to b ∈ B traverses via v . ∃A, B: A∪B. 3. Z. Applications Walter Unger 21.12.2018 12:22.
(94) Basic Definitions 1:16. Statements. Introduction to planar Graphs. Separators. 4/6. SS2016. Statements on Node-Connectivity Theorem Let G = (V , E ) with: |V | = n and |E | = m. Then is the maximal connectivity (maximal k with G is k-connected) of G : 0 falls m < n − 1 2 · m/n if m >n−1 Theorem Let G = (V , E ) connected. The following statements are equivalent: 1. v ∈ V is a node-separator.. 2. ∃a, b ∈ V : a, b 6= v : each path from a to b traverses via v . ˙ = V \ {v } and each path from a ∈ A to b ∈ B traverses via v . ∃A, B: A∪B. 3. Z. Applications Walter Unger 21.12.2018 12:22.
(95) Basic Definitions 1:16. Statements. Introduction to planar Graphs. Separators. 5/6. SS2016. Statements on Node-Connectivity Theorem Let G = (V , E ) with: |V | = n and |E | = m. Then is the maximal connectivity (maximal k with G is k-connected) of G : 0 falls m < n − 1 2 · m/n if m >n−1 Theorem Let G = (V , E ) connected. The following statements are equivalent: 1. v ∈ V is a node-separator.. 2. ∃a, b ∈ V : a, b 6= v : each path from a to b traverses via v . ˙ = V \ {v } and each path from a ∈ A to b ∈ B traverses via v . ∃A, B: A∪B. 3. Z. Applications Walter Unger 21.12.2018 12:22.
(96) Basic Definitions 1:16. Statements. Introduction to planar Graphs. Separators. 6/6. SS2016. Statements on Node-Connectivity Theorem Let G = (V , E ) with: |V | = n and |E | = m. Then is the maximal connectivity (maximal k with G is k-connected) of G : 0 falls m < n − 1 2 · m/n if m >n−1 Theorem Let G = (V , E ) connected. The following statements are equivalent: 1. v ∈ V is a node-separator.. 2. ∃a, b ∈ V : a, b 6= v : each path from a to b traverses via v . ˙ = V \ {v } and each path from a ∈ A to b ∈ B traverses via v . ∃A, B: A∪B. 3. Z. Applications Walter Unger 21.12.2018 12:22.
(97) Basic Definitions 1:17. Statements. Introduction to planar Graphs 1/5. Separators. Statements on Edge-Connectivity Theorem Let G = (V , E ) be connected. The following statements are equivalent: 1. e ∈ E is a edge-separator.. 2. e is not in any simple cycle of G .. 3. ∃a, b ∈ E : each path from a to b traverses via e. ˙ = V and each path from a ∈ A to b ∈ B traverses via e. ∃A, B: A∪B. 4. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(98) Basic Definitions 1:17. Statements. Introduction to planar Graphs 2/5. Separators. Statements on Edge-Connectivity Theorem Let G = (V , E ) be connected. The following statements are equivalent: 1. e ∈ E is a edge-separator.. 2. e is not in any simple cycle of G .. 3. ∃a, b ∈ E : each path from a to b traverses via e. ˙ = V and each path from a ∈ A to b ∈ B traverses via e. ∃A, B: A∪B. 4. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(99) Basic Definitions 1:17. Statements. Introduction to planar Graphs 3/5. Separators. Statements on Edge-Connectivity Theorem Let G = (V , E ) be connected. The following statements are equivalent: 1. e ∈ E is a edge-separator.. 2. e is not in any simple cycle of G .. 3. ∃a, b ∈ E : each path from a to b traverses via e. ˙ = V and each path from a ∈ A to b ∈ B traverses via e. ∃A, B: A∪B. 4. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(100) Basic Definitions 1:17. Statements. Introduction to planar Graphs 4/5. Separators. Statements on Edge-Connectivity Theorem Let G = (V , E ) be connected. The following statements are equivalent: 1. e ∈ E is a edge-separator.. 2. e is not in any simple cycle of G .. 3. ∃a, b ∈ E : each path from a to b traverses via e. ˙ = V and each path from a ∈ A to b ∈ B traverses via e. ∃A, B: A∪B. 4. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(101) Basic Definitions 1:17. Statements. Introduction to planar Graphs 5/5. Separators. Statements on Edge-Connectivity Theorem Let G = (V , E ) be connected. The following statements are equivalent: 1. e ∈ E is a edge-separator.. 2. e is not in any simple cycle of G .. 3. ∃a, b ∈ E : each path from a to b traverses via e. ˙ = V and each path from a ∈ A to b ∈ B traverses via e. ∃A, B: A∪B. 4. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(102) Basic Definitions 1:18. Statements. Introduction to planar Graphs. Separators. 1/4. SS2016. Definition Definition Let G = (V , E ) and (a, b) = e ∈ E . The subdivision of an edge e results in graph ˙ }, E ∪ {(a, v ), (v , b)} \ {e}) G = (V ∪{v b. a. c. Z. Applications Walter Unger 21.12.2018 12:22. d. e. Definition A set of paths of G = (V , E ) is called intern-node-disjoint, iff no two paths share an internal-node. The internal nodes are all except the start and the end node..
(103) Basic Definitions 1:18. Statements. Introduction to planar Graphs. Separators. 2/4. SS2016. Definition Definition Let G = (V , E ) and (a, b) = e ∈ E . The subdivision of an edge e results in graph ˙ }, E ∪ {(a, v ), (v , b)} \ {e}) G = (V ∪{v b. c. d. a. f. e. Z. Applications Walter Unger 21.12.2018 12:22. Definition A set of paths of G = (V , E ) is called intern-node-disjoint, iff no two paths share an internal-node. The internal nodes are all except the start and the end node..
(104) Basic Definitions 1:18. Statements. Introduction to planar Graphs. Separators. 3/4. SS2016. Definition Definition Let G = (V , E ) and (a, b) = e ∈ E . The subdivision of an edge e results in graph ˙ }, E ∪ {(a, v ), (v , b)} \ {e}) G = (V ∪{v b. c. d. a. f. e. Z. Applications Walter Unger 21.12.2018 12:22. Definition A set of paths of G = (V , E ) is called intern-node-disjoint, iff no two paths share an internal-node. The internal nodes are all except the start and the end node..
(105) Basic Definitions 1:18. Statements. Introduction to planar Graphs. Separators. 4/4. SS2016. Definition Definition Let G = (V , E ) and (a, b) = e ∈ E . The subdivision of an edge e results in graph ˙ }, E ∪ {(a, v ), (v , b)} \ {e}) G = (V ∪{v b. c. d. a. f. e. Z. Applications Walter Unger 21.12.2018 12:22. Definition A set of paths of G = (V , E ) is called intern-node-disjoint, iff no two paths share an internal-node. The internal nodes are all except the start and the end node..
(106) Basic Definitions 1:19. Statements. Introduction to planar Graphs. Separators. 1/9. Z. Applications Walter Unger 21.12.2018 12:22. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016.
(107) Basic Definitions 1:19. Statements. Introduction to planar Graphs. Separators. 2/9. Z. Applications Walter Unger 21.12.2018 12:22. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016.
(108) Basic Definitions 1:19. Statements. Introduction to planar Graphs. Separators. 3/9. Z. Applications Walter Unger 21.12.2018 12:22. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016.
(109) Basic Definitions 1:19. Statements. Introduction to planar Graphs. Separators. 4/9. Z. Applications Walter Unger 21.12.2018 12:22. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016.
(110) Basic Definitions 1:19. Statements. Introduction to planar Graphs. Separators. 5/9. Z. Applications Walter Unger 21.12.2018 12:22. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016.
(111) Basic Definitions 1:19. Statements. Introduction to planar Graphs. Separators. 6/9. Z. Applications Walter Unger 21.12.2018 12:22. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016.
(112) Basic Definitions 1:19. Statements. Introduction to planar Graphs. Separators. 7/9. Z. Applications Walter Unger 21.12.2018 12:22. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016.
(113) Basic Definitions 1:19. Statements. Introduction to planar Graphs. Separators. 8/9. Z. Applications Walter Unger 21.12.2018 12:22. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016.
(114) Basic Definitions 1:19. Statements. Introduction to planar Graphs. Separators. 9/9. Z. Applications Walter Unger 21.12.2018 12:22. Theorem Let G = (V , E ) with |V | > 3. The following statements are equivalent: 1. G is 2-connected.. 2. Each node pair is connected by two intern-node-disjoint paths.. 3. Each node pair is on a common simple cycle.. 4. There exits an edge and each node together with this edge is on a common simple cycle.. 5. There exit two edges and each pair of edges is on a common simple cycle.. 6. For each pair of nodes a, b and an edge e exists a simple path from a to b traversing e.. 7. For three nodes a, b, c exists a path from a to b traversing c.. 8. For three nodes a, b, c exists a path from a to b avoiding c.. SS2016.
(115) Basic Definitions 1:20. Statements. Introduction to planar Graphs 1/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Statements Theorem Let G = (V , E ) k-connected. Then any k nodes are on a common simple cycle. Notation: Let (G = V , E ) and (H = W , F ) graphs ˙ , E ∪ F ∪ {(a, b) | a ∈ V , b ∈ W }) G + W = (V ∪W Theorem A graph G is 3-connected, iff G may be constructed from the weel Wi = K1 + Ci (i > 4) by the following operations: 1. Adding a new edge.. 2. Splitting a node of degree > 4 into two connected nodes of degree > 3..
(116) Basic Definitions 1:20. Statements. Introduction to planar Graphs 2/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Statements Theorem Let G = (V , E ) k-connected. Then any k nodes are on a common simple cycle. Notation: Let (G = V , E ) and (H = W , F ) graphs ˙ , E ∪ F ∪ {(a, b) | a ∈ V , b ∈ W }) G + W = (V ∪W Theorem A graph G is 3-connected, iff G may be constructed from the weel Wi = K1 + Ci (i > 4) by the following operations: 1. Adding a new edge.. 2. Splitting a node of degree > 4 into two connected nodes of degree > 3..
(117) Basic Definitions 1:20. Statements. Introduction to planar Graphs 3/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Statements Theorem Let G = (V , E ) k-connected. Then any k nodes are on a common simple cycle. Notation: Let (G = V , E ) and (H = W , F ) graphs ˙ , E ∪ F ∪ {(a, b) | a ∈ V , b ∈ W }) G + W = (V ∪W Theorem A graph G is 3-connected, iff G may be constructed from the weel Wi = K1 + Ci (i > 4) by the following operations: 1. Adding a new edge.. 2. Splitting a node of degree > 4 into two connected nodes of degree > 3..
(118) Basic Definitions 1:20. Statements. Introduction to planar Graphs 4/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Statements Theorem Let G = (V , E ) k-connected. Then any k nodes are on a common simple cycle. Notation: Let (G = V , E ) and (H = W , F ) graphs ˙ , E ∪ F ∪ {(a, b) | a ∈ V , b ∈ W }) G + W = (V ∪W Theorem A graph G is 3-connected, iff G may be constructed from the weel Wi = K1 + Ci (i > 4) by the following operations: 1. Adding a new edge.. 2. Splitting a node of degree > 4 into two connected nodes of degree > 3..
(119) Basic Definitions 1:20. Statements. Introduction to planar Graphs 5/5. Separators. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Statements Theorem Let G = (V , E ) k-connected. Then any k nodes are on a common simple cycle. Notation: Let (G = V , E ) and (H = W , F ) graphs ˙ , E ∪ F ∪ {(a, b) | a ∈ V , b ∈ W }) G + W = (V ∪W Theorem A graph G is 3-connected, iff G may be constructed from the weel Wi = K1 + Ci (i > 4) by the following operations: 1. Adding a new edge.. 2. Splitting a node of degree > 4 into two connected nodes of degree > 3..
(120) Basic Definitions 1:21. Statements. Introduction to planar Graphs 1/3. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Statements on k-Connectivity Theorem (Menger’s Theorem) G is k-connected, iff any two node are connected by k intern-node-disjoint paths. Theorem (Menger’s Theorem) G is k-edge-connected, iff any two node are connected by k edge-disjoint paths.. SS2016.
(121) Basic Definitions 1:21. Statements. Introduction to planar Graphs 2/3. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Statements on k-Connectivity Theorem (Menger’s Theorem) G is k-connected, iff any two node are connected by k intern-node-disjoint paths. Theorem (Menger’s Theorem) G is k-edge-connected, iff any two node are connected by k edge-disjoint paths.. SS2016.
(122) Basic Definitions 1:21. Statements. Introduction to planar Graphs 3/3. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Statements on k-Connectivity Theorem (Menger’s Theorem) G is k-connected, iff any two node are connected by k intern-node-disjoint paths. Theorem (Menger’s Theorem) G is k-edge-connected, iff any two node are connected by k edge-disjoint paths.. SS2016.
(123) Basic Definitions 1:22. Statements. Introduction to planar Graphs. Separators. 1/9. Z. Applications Walter Unger 21.12.2018 12:22. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016.
(124) Basic Definitions 1:22. Statements. Introduction to planar Graphs. Separators. 2/9. Z. Applications Walter Unger 21.12.2018 12:22. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016.
(125) Basic Definitions 1:22. Statements. Introduction to planar Graphs. Separators. 3/9. Z. Applications Walter Unger 21.12.2018 12:22. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016.
(126) Basic Definitions 1:22. Statements. Introduction to planar Graphs. Separators. 4/9. Z. Applications Walter Unger 21.12.2018 12:22. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016.
(127) Basic Definitions 1:22. Statements. Introduction to planar Graphs. Separators. 5/9. Z. Applications Walter Unger 21.12.2018 12:22. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016.
(128) Basic Definitions 1:22. Statements. Introduction to planar Graphs. Separators. 6/9. Z. Applications Walter Unger 21.12.2018 12:22. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016.
(129) Basic Definitions 1:22. Statements. Introduction to planar Graphs. Separators. 7/9. Z. Applications Walter Unger 21.12.2018 12:22. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016.
(130) Basic Definitions 1:22. Statements. Introduction to planar Graphs. Separators. 8/9. Z. Applications Walter Unger 21.12.2018 12:22. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016.
(131) Basic Definitions 1:22. Statements. Introduction to planar Graphs. Separators. 9/9. Z. Applications Walter Unger 21.12.2018 12:22. Computing the Connectivity Theorem The 1-connectivity of a graph may be computed by DFS/BFS. Theorem The 1-edge-connectivity of a graph may be computed by DFS/BFS. Theorem The 2-connectivity of a graph may be computed by DFS/BFS. Theorem The k-connectivity of a graph may be computed by flow algorithms. Theorem The k-edge-connectivity of a graph may be computed by flow algorithms.. SS2016.
(132) Basic Definitions 1:23. Definitions. Introduction to planar Graphs 1/3. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Definitions Definition A graph G = (V , E ) is called planar, iff it could be drawn in the plane without crossing edges. A connected area of such an embedding is called window. The unlimited window is called outer window. Definition A graph G = (V , E ) is called maximal planar, iff the adding of an edge makes G non-planar.. SS2016.
(133) Basic Definitions 1:23. Definitions. Introduction to planar Graphs 2/3. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Definitions Definition A graph G = (V , E ) is called planar, iff it could be drawn in the plane without crossing edges. A connected area of such an embedding is called window. The unlimited window is called outer window. Definition A graph G = (V , E ) is called maximal planar, iff the adding of an edge makes G non-planar.. SS2016.
(134) Basic Definitions 1:23. Definitions. Introduction to planar Graphs 3/3. Separators. Z. Applications Walter Unger 21.12.2018 12:22. Definitions Definition A graph G = (V , E ) is called planar, iff it could be drawn in the plane without crossing edges. A connected area of such an embedding is called window. The unlimited window is called outer window. Definition A graph G = (V , E ) is called maximal planar, iff the adding of an edge makes G non-planar.. SS2016.
(135) Basic Definitions 1:24. Definitions. Introduction to planar Graphs. Separators. 1/4. Example: planar Graph v0. v1. v5. v3. v6. v4. v2. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(136) Basic Definitions 1:24. Definitions. Introduction to planar Graphs. Separators. 2/4. Example: planar Graph v0. v1. v5. v3. v6. v4. v2. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(137) Basic Definitions 1:24. Definitions. Introduction to planar Graphs. Separators. 3/4. Example: planar Graph v0. v1. v5. v3. v6. v4. v2. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(138) Basic Definitions 1:24. Definitions. Introduction to planar Graphs. Separators. 4/4. Example: planar Graph v0. v1. v5. v3. v6. v4. v2. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(139) Basic Definitions 1:25. Theorems on planar Graphs. Introduction to planar Graphs 1/3. Separators. Results I. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Theorem If G = (V , E ) is planar and 2-connected, then each window is a simple cycle and each edge separates two different windows. Theorem (Euler) Let G = (V , E ) be a planar graph with |V | = n, |E | = m. Let f be the number of windows and k be the number of connected components. Then the following holds: n − m + f = 1 + k. Proof by simple induction..
(140) Basic Definitions 1:25. Theorems on planar Graphs. Introduction to planar Graphs 2/3. Separators. Results I. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Theorem If G = (V , E ) is planar and 2-connected, then each window is a simple cycle and each edge separates two different windows. Theorem (Euler) Let G = (V , E ) be a planar graph with |V | = n, |E | = m. Let f be the number of windows and k be the number of connected components. Then the following holds: n − m + f = 1 + k. Proof by simple induction..
(141) Basic Definitions 1:25. Theorems on planar Graphs. Introduction to planar Graphs 3/3. Separators. Results I. Z. Applications Walter Unger 21.12.2018 12:22. SS2016. Theorem If G = (V , E ) is planar and 2-connected, then each window is a simple cycle and each edge separates two different windows. Theorem (Euler) Let G = (V , E ) be a planar graph with |V | = n, |E | = m. Let f be the number of windows and k be the number of connected components. Then the following holds: n − m + f = 1 + k. Proof by simple induction..
(142) Basic Definitions 1:26. Theorems on planar Graphs. Introduction to planar Graphs. Separators. 1/13. Proof n − m + f = 1 + k holds for a single node. new node: (n + 1) − m + f = 1 + (k + 1) new edge connects components: n − (m + 1) + f = 1 + (k − 1) or new edge seperates window: n − (m + 1) + (f + 1) = 1 + k.. a1. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(143) Basic Definitions 1:26. Theorems on planar Graphs. Introduction to planar Graphs. Separators. 2/13. Proof n − m + f = 1 + k holds for a single node. new node: (n + 1) − m + f = 1 + (k + 1) new edge connects components: n − (m + 1) + f = 1 + (k − 1) or new edge seperates window: n − (m + 1) + (f + 1) = 1 + k.. a1. c1. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(144) Basic Definitions 1:26. Theorems on planar Graphs. Introduction to planar Graphs. Separators. 3/13. Proof c2. n − m + f = 1 + k holds for a single node. new node: (n + 1) − m + f = 1 + (k + 1) new edge connects components: n − (m + 1) + f = 1 + (k − 1) or new edge seperates window: n − (m + 1) + (f + 1) = 1 + k.. a1. c1. Z. Applications Walter Unger 21.12.2018 12:22. SS2016.
(145) Basic Definitions 1:26. Theorems on planar Graphs. Introduction to planar Graphs. Separators. 4/13. Proof c2. n − m + f = 1 + k holds for a single node.. o2. new node: (n + 1) − m + f = 1 + (k + 1) new edge connects components: n − (m + 1) + f = 1 + (k − 1) or new edge seperates window: n − (m + 1) + (f + 1) = 1 + k.. a1. Z. Applications Walter Unger 21.12.2018 12:22. r1. e2. o1 c1. e1. SS2016.
(146) Basic Definitions 1:26. Theorems on planar Graphs. Introduction to planar Graphs. Separators. 5/13. Proof c2. n − m + f = 1 + k holds for a single node.. o2. new node: (n + 1) − m + f = 1 + (k + 1) new edge connects components: n − (m + 1) + f = 1 + (k − 1) or new edge seperates window: n − (m + 1) + (f + 1) = 1 + k.. a1. Z. Applications Walter Unger 21.12.2018 12:22. r1. e2. o1 c1. e1. SS2016.
(147) Basic Definitions 1:26. Theorems on planar Graphs. Introduction to planar Graphs. Separators. 6/13. Proof c2. n − m + f = 1 + k holds for a single node.. o2. new node: (n + 1) − m + f = 1 + (k + 1) new edge connects components: n − (m + 1) + f = 1 + (k − 1) or new edge seperates window: n − (m + 1) + (f + 1) = 1 + k.. a1. Z. Applications Walter Unger 21.12.2018 12:22. r1. e2. o1 c1. e1. SS2016.
(148) Basic Definitions 1:26. Theorems on planar Graphs. Introduction to planar Graphs. Separators. 7/13. Proof c2. n − m + f = 1 + k holds for a single node.. o2. new node: (n + 1) − m + f = 1 + (k + 1) new edge connects components: n − (m + 1) + f = 1 + (k − 1) or new edge seperates window: n − (m + 1) + (f + 1) = 1 + k.. a1. Z. Applications Walter Unger 21.12.2018 12:22. r1. e2. o1 c1. e1. SS2016.
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Colouring Problem on Permutation-Graphs Theorem The graph-to-colour problem is sovable in time On logn on permutation-graphs.. Idea: Analog algorithms as
Motivation Till now: Problems are efficient solvable, if the “flow of information is not too large”.. Example: interval-graphs, permutation-graphs,
Example Independet Set Thus we only have to define the following: What we store for f pi : Df pi The procedure Df p1 := Initf p1 , to compute the initial values The procedure Df pi+1
Planar graphs: no Intervall-graphs: yes Arc-graphs: no Permutation-graphs: yes Outerplanar graphs: no Maximal outerplanar graphs: yes Maximal planar graphs: no following slide
1-Way Gossip on Cycles Idea Messages should traverse in both directions.. Activate each f n-th node on
Gossip on arbitrary Trees Proof III Consider the two-way mode: by a similar way we may prove: At time t only two neighbours nodes u and v know the total information.. We get in
