also exist for the general case of AT-free graphs. However, none of the techniques used for the (bad-)claw-free graphs can be transferred. Corneil and Stacho [38] already showed that there are AT-free graphs which do not possess AT-free orders that are also LBFS orders. In addition, Figure 2.10 shows a graph which does not possess a bilateral AT-free ordering. Therefore, it will be necessary to use a different search algorithm, possibly a BFS-derivative based on BFSconv. We summarise these suppositions in the following conjecture.
Conjecture 2.3.12. Let G= (V, E) be an AT-free graph. There is a linear time algo-rithm that computes an AT-free (BFS) order.
Chordal graphs are well-known to possess many good structural and algorithmic prop-erties [17, 49, 64, 75]. The main goal of this chapter is to study certain concepts related to chordal graphs in the framework of more general graph classes. The starting point for our research is a result due to Dirac [49] stating that every minimal cutset in a chordal graph is a clique, which implies that every chordal graph with at least one vertex has a simplicial vertex, that is, a vertex whose neighbourhood is a clique [64]. Denoting byPk thek-vertex path, this result can be formulated as follows.
Theorem 3.0.1. Every chordal graph with at least one vertex has a vertex v such that v is not the midpoint of any induced P3.
This theorem was generalised in the literature in various ways, see, e.g., [35, 93, 106, 117, 133]. Two particular ways of generalising Theorem 3.0.1 include:
(i) proving a property of general graphs that, when specialized to chordal graphs, results in the existence of a simplicial vertex, and
(ii) generalising the ‘simpliciality’ property from vertices, which are paths of length 0, to longer induced paths, and proving the existence of such paths for graphs excluding suitably longer cycles.
Let us explain in more detail the corresponding results.
First generalisation – from chordal graphs to all graphs
A generalisation of the first kind is given by the following theorem, which follows from [117, Theorem 3] as well as from [13, Main Theorem 4.1] and [1, Lemma 2.3].
Theorem 3.0.2. Every graph G with at least one vertex has a vertexv such that every induced P3 having v as its midpoint is contained in an induced cycle in G.
The above property of vertices will be one of the central concepts for this chapter and we formalize it as follows.
Definition 3.0.3. A vertexv in a graph Gis said to be avoidableif between any pairx andy of neighbours of v there exists anx, y-path, all the internal vertices of which avoid v and all neighbours of v. Equivalently, a vertex v is avoidable if every inducedP3 with midpoint v closes to an induced cycle.
This terminology is motivated by considering a setting whereGrepresents a symmetric acquaintance relation on a group of people. In this setting, the property of person (equivalently, vertex) a being avoidable can be interpreted as follows: whenever two acquaintances of a need to share some information that they would not like to share witha, they can do so by passing the information along a path completely avoiding both aand all her other acquaintances. Thus, a is in a sense avoidable, as information can be passed around in her immediate proximity without her knowledge.
Note that every simplicial vertex in a graph is avoidable. If we analyse avoidable vertices in graph classes, rather than in general graphs, we see that this definition is a generalisation of many well known concepts. For example, in a tree a vertex is avoidable if and only if it is a leaf, while in a chordal graph a vertex is avoidable if and only if it is simplicial. With this terminology, Theorem 3.0.2 can be equivalently stated as follows.
Theorem 3.0.4. Every graph with at least one vertex has an avoidable vertex.
The notion of avoidable vertices has appeared in the literature (with different termi-nology) in a variety of settings. To our knowledge, the earliest appearance was in the paper from 1976 by Ohtsuki et al. [117], where avoidable vertices were characterized as exactly the vertices from which a minimal elimination ordering can start. Here, a min-imal elimination ordering of a graph G = (V, E) is a procedure of eliminating vertices one at a time so that before each vertex is removed, its neighbourhood is turned into a clique, and the resulting setF of edges added throughout the procedure is an inclusion-minimal set of non-edges ofGsuch that (V, E∪F) is a chordal graph (in other words, (V, E∪F) is aminimal triangulation of G). Given a graphG, an avoidable vertex in G can be found in linear time using graph search algorithms such as Lexicographic Breadth First Search (LBFS) [126] (see also [79]) or Maximum Cardinality Search (MCS) [14].
The presentation closest to our setting is the one used by Ohtsuki et al. [117]. In fact, Berry et al. [14, 15] named avoidable verticesOCF-vertices, after the initials of the three authors of [117].
Second generalisation – from vertices to longer paths
In order to generalize the notion of simplicial vertices to longer paths, the next definition, partially following Chv´atal et al. [35], will be useful.
Definition 3.0.5. Given an induced path P in a graph G, a two-sided extension of P is any induced path inGobtained by adding toP one edge at each end. An induced path is said to be simplicial if it has no two-sided extension.
In this terminology, Theorem 3.0.1 can be stated as follows: every graph with at least one vertex and without induced cycles of length more than 3 has a simplicial induced P1. Chv´atal et al. [35] generalized this result as follows.
Theorem 3.0.6(Chv´atal et al. [35]). For eachk≥1, every {Ck+3, Ck+4, . . .}-free graph that contains an inducedPk also contains a simplicial induced Pk.
A common generalisation?
Theorems 3.0.4 and 3.0.6 suggest that a further common generalisation might be possible, based on the following generalisation of Definition 3.0.3 (definition of avoidable vertices) to longer paths.
Definition 3.0.7. An induced path P in a graph G is said to be avoidable if every two-sided extension of P is contained in an induced cycle.
Thus, in particular, a vertexvin a graphGis avoidable if and only if the corresponding one-vertex path is avoidable. Moreover, every simplicial induced path is (vacuously) avoidable.
We conjecture that the following common generalisation of Theorems 3.0.4 and 3.0.6 holds.
Conjecture 3.0.8. For every k ≥ 1, every graph that contains an induced Pk also contains an avoidable induced Pk.
Theorem 3.0.4 implies the conjecture fork= 1, while Theorem 3.0.6 implies it for every positive integer k, provided we restrict ourselves to the class of graphs without induced cycles of length more than k+ 2. Indeed, if G is a {Ck+3, Ck+4, . . .}-free graph that contains an induced Pk, then by Theorem 3.0.6 graph G contains a simplicial induced Pk, and every simplicial induced path is avoidable.
The results given in this chapter are joint work with Maria Chudnovsky, Vladimir Gur-vich, Martin Milaniˇc and Mary Servatius. A published extended abstract of this work can be found in [7].
3.1 Characterisation and Existence of Avoidable Vertices
The proof of Theorem 3 in the paper [117] by Ohtsuki, Cheung, and Fujisawa (which itself relied on earlier works of Rose [123, 124, 125]) leads to the characterisation of avoidable vertices given by the following theorem. Since we are not aware of any explicit statement of this result in the literature, we state it here and give a short self-contained proof that does not rely on the concept of minimal elimination orderings.
Theorem 3.1.1. Let G= (V, E) be a graph and letv∈V. Then v is avoidable in Gif and only if v is a simplicial vertex in some minimal triangulation of G.
Proof. LetG0 = (V, E∪F) be a minimal triangulation ofGand letv∈V be a simplicial vertex inG0. Suppose for a contradiction that vis not avoidable in G. Then, v contains two neighbours, sayxandy, such thatxandybelong to different connected components of the graph G−S, where S = NG[v]\ {x, y}. Since v is simplicial in G0, set S is a clique inG0. LetF∗ be the set of all pairs{u, w} ∈F such thatuandw are in different connected components of the graph G−S and let G∗ be the graph (V, E∪(F \F∗)).
Since S is a clique in G∗, no induced cycle of G∗ contains vertices from two different components ofG∗−S. It follows that every induced cycle inG∗ is also an induced cycle
inG0, and the fact thatG0 is chordal implies that G∗ is chordal. However, since the set F∗ is non-empty (note that it contains {x, y}), the fact that G∗ is chordal contradicts the assumption that G0 = (V, E∪F) is a minimal triangulation of G. This shows that v is avoidable inG.
For the converse direction, letv∈V be an avoidable vertex inG and letS =NG(v).
Let F0 denote the set of non-adjacent vertex pairs in S and let G0 = (V0, E0) be the graph obtained from G−v by turning S into a clique (that is, V0 = V \ {v} and E0 =E(G−v)∪F0). Moreover, let G01 = (V0, E0 ∪F0) be a minimal triangulation of G0 and let G1 = (V, E∪(F0∪F0)). Note that S is a clique in G0 and therefore also in G01. Since G1 can be obtained from the chordal graph G01 by adding to it vertex v and making it adjacent to all vertices of clique S, graph G1 is chordal. Furthermore, since v is a simplicial vertex in G1, to complete the proof it suffices to show that G1
is a minimal triangulation of G, or, equivalently, that for every edge f ∈ F0 ∪F0 the graph G1 −f is not chordal. Suppose first that f ∈ F0. Since G01 = (V0, E0 ∪F0) is a minimal triangulation of G0, the graph G01 −f is not chordal, and thus it contains an induced cycle C of length at least 4. As G01 = G1−v, we see that C is also an induced cycle in G1−f. Finally, suppose that f ∈ F0. Then f = {x, y} where x and y are two non-adjacent neighbours of v in G. Let S0 = NG[v]\ {x, y}. Since v is an avoidable vertex inG, verticesxand y are in the same component of the graphG−S0, and consequently, sinceGis a spanning subgraph ofG1−f, also in the same component of the graph (G1−f)−S0. LetP be a shortestx, y-path (G1−f)−S0. ThenV(P)∪ {v}
induces a cycle of length at least four in the graphG1−f, which implies thatG1−f is not chordal, as claimed. This completes the proof.
Remark 3.1.2. Sets of vertices of a graph Gthat are maximal cliques in some minimal triangulation of G were studied in the literature under the name potential maximal cliques. This concept was introduced by Bouchitt´e and Todinca in [19] and has already found many applications in algorithmic graph theory (see, e.g., [20, 61, 62]). In this terminology, Theorem 3.1.1 states that given a vertexv∈V(G), its closed neighbourhood NG[v] is a potential maximal clique in G if and only if v is avoidable.
Since every graph has a minimal triangulation, Theorems 3.0.1 and 3.1.1 imply The-orem 3.0.4. An application of TheThe-orem 3.0.4 to vertex-transitive graphs will be given in Section 3.5.
We now discuss the consequence of Theorem 3.0.4 when the theorem is applied to the line graph of a given graph. An edgeein a graphGis said to bepseudo-avoidable if the corresponding vertex in the line graph L(G) is avoidable. It is not difficult to see that an edgeein a graph G is pseudo-avoidable if and only if any (not necessarily induced) 3-edge path having e as the middle edge closes to a (not necessarily induced) cycle in G. Note that the concepts of avoidable edges (considered as inducedP2s, in the sense of Definition 3.0.7) and of pseudo-avoidable edges are incomparable, see Fig. 3.1. Assuming notation from Fig. 3.1, one can see thateis not an avoidable edge in G. However, it is pseudo-avoidable, aseis an avoidable vertex inL(G). On the other hand,f is avoidable (even simplicial) inG. However, it is not pseudo-avoidable.
Applying Theorem 3.0.4 to the line graph of the given graph yields the following.
G:
a b
c d
e f
g L(G):
a d f b
c g
e Figure 3.1: A graphG and its line graphL(G).
Corollary 3.1.3. Every graph with an edge has a pseudo-avoidable edge.
An application of Corollary 3.1.3 to edge-transitive graphs will be given in Section 3.5.
Recall that Theorem 3.0.4 coincides with the statement of Conjecture 3.0.8 for the casek= 1. We now reprove this statement in a slightly stronger form, using an approach that we will adapt in Section 3.4 for the proof of the case k = 2 of the conjecture. A vertex in a graph is said to be universal if it is adjacent to every other vertex, and non-universal otherwise.
Theorem 3.1.4. For every graph G and every non-universal vertex v ∈ V(G) there exists an avoidable vertex a∈V(G)\N[v].
Proof. Suppose that the theorem is false and take a counterexampleGwith the smallest possible number of vertices and, subject to that, with the largest possible number of edges. The minimality of |V(G)| implies that G is connected. Since G is a counterex-ample, it has a non-universal vertex v∈V(G) such that no vertex not adjacent tov is avoidable.
Letb ∈V(G)\N[v]. Since b is not avoidable in G, it is the midpoint of an induced P3, say x−b−y, that is not contained in any induced cycle. Observe that vertices x and y are in different components of the graph G−(N[b]\ {x, y}), since any induced path P from x to y in G−(N[b]\ {x, y}) would imply the existence of an induced cycle x−b−y−P−x closing the 3-vertex path x−b−y. It follows that the graph G−(N[b]\{x, y}) is disconnected. In particular, there exists an inclusion-minimal subset S ofN(b) such thatG−(S∪ {b}) is disconnected. Note that v6∈N[b], and, hence,v is a vertex of G−(S∪ {b}). Let C be the component of G−(S∪ {b}) containingv. It follows from the minimality ofS that every vertex inS has a neighbour in C.
Suppose first thatbis not universal inG−C. The minimality of |V(G)|implies that G−C is not a counterexample to the theorem. Therefore, there exists an avoidable vertex ain the graph G−C that is neither equal nor adjacent to b. Since ais a vertex of G−(S∪ {b}) not contained in C, it is not adjacent to v. We claim that a is also avoidable in G, which will contradict the assumption that no vertex not adjacent to v in Gis avoidable. Let P be an induced P3 inG with midpoint a. Sincea belongs to a component ofG−(S∪ {b}) different fromC, no vertex ofC is adjacent toa. Therefore, P is an induced P3 in the graph G−C. Since a is avoidable in G−C, there exists an induced cycle in G−C containing P. Since G−C is an induced subgraph of G, we
conclude that there exists an induced cycle inG containing P. Since P was arbitrary, it follows thatais avoidable in G; a contradiction.
Therefore, we may assume thatbis universal inG−C. LetG0 be the graph obtained fromG−C by adding a new vertexdand makingdadjacent to every vertex inS∪ {b}.
Note that ifC6={v}, then G0 has strictly fewer vertices than G. If, on the other hand, C ={v}, then G0 has the same number of vertices as G but strictly more edges, since NG(C)⊆S, whileNG0(d) =S∪ {b}. Denoting by C0 any component of G−(S∪ {b}) other thanC, we see that every vertex of C0 is not adjacent todinG0. Hence,dis not universal inG0 and the choice ofGimplies thatG0 has an avoidable vertexathat is not dand not adjacent tod. This shows that a∈V(G)\(C∪S∪ {b}).
We claim that a is avoidable in G. Suppose that x, y ∈ NG(v) are not adjacent.
We need to show that the path x−a−y closes to an induced cycle. We have x, y ∈ NG0(a)\ {b}. Sinceais avoidable inG0, there exists an induced path P fromxtoyinG0 such thatahas no neighbours inV(P)\ {x, y}. Ifd6∈V(P), thena−x−P−y−ais the required cycle inG. Therefore, we may assume that d∈V(P). Observe thatd6=x, y.
Letpand q be the two neighbours of dinV(P). Then p, q∈V(G). Since b is universal inG0 andb6=a, it follows thatb6∈V(P). Since pandq are adjacent todandb6∈V(P), it follows thatp, q∈S. Moreover, notice thatV(P)\ {p, q}is disjoint fromS∪ {b}. By the minimality ofS and, since C is connected, there is an induced path Q inGfrom p toq such thatV(Q)\ {p, q} ⊆C. However, in this casea−x−P−p−Q−q−P−y−a is the required cycle in G. This shows that ais an avoidable vertex in G not adjacent tov. This contradicts the assumption onv and completes the proof of the theorem.