MICHAEL R. FELLOWS†, FRANCES A. ROSAMOND†, UDI ROTICS‡, AND STEFAN SZEIDER§
Abstract. Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic with second-order quantification on vertex sets, that includes NP-hard problems such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width. We show that the clique-width of a given graph cannot be absolutely approximated in polynomial time unless P = NP. We also show that, given a graphGand an integerk, deciding whether the clique-width ofGis at mostk is NP-complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s.
Key words.clique-width, NP-completeness, pathwidth, absolute approximation AMS subject classifications.68Q17, 05C75, 68Q42
1. Introduction. Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. This parameter is defined via a graph construction process where only a limited number of vertex labels are available; vertices that share the same label at a certain point of the construction process must be treated uniformly in subsequent steps. In particular, one can use the following four operations: the creation of a new vertex with labeli, the vertex-disjoint union of already constructed labeled graphs, the insertion of all possible edges between vertices of specified labels, and the uniform relabeling of vertices. The clique-width cwd(G) of a graph G is the smallest number k of labels that suffice to constructG by means of these four operations. Such a construction of a graph can be represented by an algebraic term called ak-expression. More exact definitions are provided in Section 2.
This composition mechanism was first considered by Courcelle, Engelfriet, and Rozenberg [9, 10]. Clique-width can be considered to be more general than the pop- ular graph parameter treewidth since there are graphs of constant clique-width but arbitrarily high treewidth (e.g., complete graphs), but graphs of bounded treewidth also have bounded clique-width [14, 7]. In recent years, clique-width has received much attention [4, 5, 3, 8, 6, 13, 11, 12, 14, 15, 7, 16, 19, 22, 26, 28, 27, 33].
In particular, the following result of Courcelle, Makowsky, and Rotics [11] makes the parameter clique-width attractive: any graph problem that can be expressed in Monadic Second Order Logic with second-order quantification on vertex sets (MSO1) can be solved in linear time for graphs of clique-width bounded by some constantk;
albeit the running time involves a constant which can be multiply exponential ink.
A k-expression must be provided as input to the algorithm. Many NP-hard graph problems, e.g., 3-colorability, can be expressed in MSO1. Seese [32] conjectured that if Cis a class of graphs such that it is decidable for every property that can be expressed in MSO1 whether there exists a graph in C that satisfies the property, then C is of
∗An extended abstract of this paper appeared in the Proceedings of STOC 2006, 38th ACM Symposium on Theory of Computing, Seattle, Washington, USA, pp. 354–362, ACM Press, 2006.
†School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan 2308 NSW, Australia ([mike.fellows|frances.rosamond]@cs.newcastle.edu.au).
‡School of Computer Science and Mathematics, Netanya Academic College, Netanya, Israel (rotics@mars.netanya.ac.il).
§Department of Computer Science, Durham University, Durham, England, UK (stefan.szeider@durham.ac.uk).
bounded clique-width. This conjecture can be considered as a kind of converse to the result of [11]. A slightly weaker form of Seese’s Conjecture, where MSO1with parity predicates is considered, was recently shown by Courcelle and Oum [15].
A main limit for applications of the result of Courcelle, et al. [11] is that it is not known how to efficiently obtaink-expressions for graphs of clique-widthk(except for graphs belonging to special classes [29]). In particular, the following question has been open since the introduction of clique-width in the early 1990s.
Question 1. Is it possible to compute the clique-width of a graph in polynomial time?
Since the computation of treewidth is well-known to be NP-hard (Arnborg, Corneil, and Proskurowski [1]), it is obvious to assume that such a hardness result should also hold for the more general parameter clique-width. Despite considerable ef- forts, no hardness result for clique-width has until now been obtained. A main obstacle for giving a reduction from a known NP-hard problem is certainly the strong gener- ative power of clique-width: even very few labels are sufficient to construct graphs of high connectivity; e.g., two labels are sufficient to construct complete graphs of arbitrary size (see below).
In the present paper we answer Question 1 negatively as follows.
Theorem 1. Let ε be a constant with 0 ≤ ε < 1. The clique-width of graphs withn vertices of degree greater than 2 cannot be approximated by a polynomial-time algorithm with an absolute error guarantee ofnε unlessP = NP.
In particular, there is no polynomial-time absolute approximation algorithm for clique-width unlessP = NP.
Theorem 2. cwd-recognitionisNP-complete.
Here, cwd-recognitionrefers to the following decision problem: given a graph G and an integer k, is the clique-width of G at most k? We shall use a similar terminology for other graph parameters such as pathwidth (pwd).
Our proofs rely ultimately on a reduction from pwd-recognition. However, a direct reduction, where an instance (G, k) of pwd-recognitionis reduced to an instance (G", k") of cwd-recognition, seems to be difficult to obtain, as the combina- torics for such a construction would be obliged to be “tight.” We base our reduction on the following stronger result of Bodlaender, Gilbert, Hafsteinsson, and Kloks [2]: It is NP-hard to approximate pathwidth within an absolute error guarantee. This stronger basis allows the reduction to be lax, making the combinatorial effort somehow easier than it would be for a direct reduction.
Obtaining the hardness for the exact solution of one problem from the non- approximability of another problem is the key for our success with respect to Ques- tion 1. We believe that this approach could be useful in many situations, possibly also in parameterized complexity.
1.1. Related Work. Corneil, Habib, Lanlignel, Reed, and Rotics [6] show that graphs of clique-width at most 3 can be recognized in polynomial time. Oum and Sey- mour [31] present an algorithm that, for fixedk, computes (23k+2−1)-expressions for n-vertex graphs of clique-width at mostkin timeO(n9logn). This result renders the notion “class of bounded clique-width” feasible in much the same way that “class of bounded treewidth” is feasible. The algorithm of Oum and Seymour computes first an approximate “rank-decomposition” (a decomposition associated with the graph invari- ant rank-width, for definitions see [31, 30]) and then converts the rank-decomposition of width k into a (2k−1 + 1)-expressions. Oum [30] gives improved algorithms for rank-width approximation that also give rise to improved algorithms for clique-width
(however, still with an approximation error that is exponential in the clique-width).
The graph parameter “NLC-width” introduced by Wanke [34] is defined simi- larly to clique-width using a single type of operation that combines disjoint union and insertion of edges. Gurski and Wanke [20] show that NLC-width-recognition is NP-hard. Since NLC-width and clique-width can differ by a factor of 2 (see Jo- hansson [23]), non-approximability with an absolute error guarantee (or NP-hardness of recognition) for one of the two parameters does not imply a similar result for the other parameter.
2. Definitions and preliminaries. Unless otherwise stated, all graphs consid- ered in this paper are finite, undirected, and simple.
2.1. Clique-width. Let k be a positive integer. A k-graph is a graph whose vertices are labeled by integers from{1, . . . , k}. We consider an arbitrary graph as a k-graph with all vertices labeled by 1. We call the k-graph consisting of exactly one vertexv (say, labeled byi∈ {1, . . . , k}) aninitial k-graph and denote it byi(v).
Theclique-width cwd(G) of a graphGis the smallest integer ksuch thatGcan be constructed from initialk-graphs by means of repeated application of the following three operations.
1. Disjoint union (denoted by⊕);
2. Relabeling: changing all labelsitoj (denoted byρi→j);
3. Edge insertion: connecting all vertices labeled byi with all vertices labeled byj,i%=j (denoted byηi,j or ηj,i); already existing edges are not doubled.
A construction of a k-graph using the above operations can be represented by an algebraic term composed of ⊕, ρi→j, andηi,j, (i, j ∈ {1, . . . , k}, andi %= j). Such a term is called a cwd-expression defining G. Ak-expression is a cwd-expression in which at most k different labels occur. Thus, the clique-width of a graph G is the smallest integerksuch thatGcan be defined by ak-expression.
For example, the complete graphK4 on the verticesu,v,w, xis defined by the cwd-expression
ρ2→1(η1,2(ρ2→1(η1,2(ρ2→1(η1,2(2(u)⊕1(v)))⊕2(w)))⊕2(x))).
Hence cwd(K4)≤2. In general, every complete graph on n≥2 vertices has clique- width exactly 2.
2.2. Directed clique-width. One can use ak-expression to construct a directed graph, interpretingηi,j as the operation that inserts directed edges that are oriented from vertices labeled i to vertices labeled j. Courcelle and Olariu [14] define the clique-width of a directed graph D as the smallest integer k such that D can be constructed by ak-expression. Theorems 1 and 2 carry over to directed graphs; this follows from the following considerations.
LetGbe an undirected graph, and letD be the directed graph obtained fromG by replacing every undirected edgeuvofGwith two directed edges (u, v) and (v, u).
It is easy to see that cwd(G) = cwd(D) since from anyk-expression forGwe can build ak-expression forDby adding anηi,j-operation immediately after anyηj,i-operation.
Thus, Theorems 1 and 2 imply their counterparts for directed graphs.
2.3. Linear clique-width. A cwd-expression islinear if whenever twok-graphs are put together by disjoint union, at least one of the two graphs is an initialk-graph (in contrast to non-linear cwd-expressions where both k-graphs can be arbitrarily large). In other words, a linear k-expression encodes the composition of a graph
G G2 G! G!!
Fig. 3.1.Illustration for Constructions 1, 2, and 3.
starting with a single vertex to which one adds successively further vertices, one after the other, with interleaved operations of relabeling and edge insertion. The parse trees of lineark-expressions are path-like; hence one can consider the relation between linear clique-width and clique-width as analogous to the relation between pathwidth and treewidth. Thelinear clique-widthlin-cwd(G) of a graphGis the smallest integer ksuch thatGcan be defined by a lineark-expression.
Every complete graph onn≥2 vertices has linear clique-width exactly 2 (observe that the cwd-expression for K4 as given above is linear). However, the difference between the clique-width and the linear clique-width of a graph can be arbitrarily large. LetThdenote a complete ternary tree of heighthand letTh" denote the graph obtained fromTh by means of Construction 2 as defined below. In [17] we show that the clique-width ofTh" is at most 4, but the linear clique-width ofTh" is at leasth−1.
2.4. Treewidth and pathwidth. LetTbe a tree andχa labeling of the vertices ofT by sets of vertices of a graphG. The pair (T, χ) is atree decomposition of Gif (i) every vertex ofG belongs to χ(t) for some vertext of T; (ii) for every edge vw ofGthere is some vertex tof T withv, w∈χ(t); (iii) for any verticest1, t2, t3 of T, ift2 lies on a path from t1 to t3, then χ(t1)∩χ(t3)⊆χ(t2). The width of (T, χ) is the maximum |χ(t)| −1 over all verticest of T. Thetreewidth twd(G) of G is the minimum width over all tree-decompositions of G. The pathwidth pwd(G) of Gis the minimum width over all tree-decompositions (T, χ) ofGwhere T is a path.
2.5. Cobipartite graphs. A graph is cobipartite if it is the complement of a bipartite graph. That is, the vertex set of a cobipartite graphGcan be partitioned into two sets A and B such that each of them induces in G a complete subgraph.
Several graph parameters, including treewidth and pathwidth, agree on cobipartite graphs (see, e.g., Fomin, Heggernes, and Telle [18]).
3. Proof of the main results. In what follows, letαdenote an integer-valued graph parameter. We consider the following decision problem.
α-recognition
Instance: A graphGand a positive integerk.
Question: Isα(G) at mostk?
Arnborg et al. [1] have shown that pwd-recognitionis NP-complete, even for cobi- partite graphs.
The following construction is due to Bodlaender, Gilbert, Hafsteinsson, and Kloks [2] (this and other constructions defined below are illustrated in Fig. 3.1).
Construction 1. Given a graphG, and an integerq≥1, we obtain a graphGq by replacing each vertexv ofG byq verticesv1, . . . , vq, and by joining two vertices vi, wj by an edge if eitherv=wandi%=j, orv%=w andvw is an edge ofG.
Note that if G is cobipartite then so is Gq (as pointed out by Karpinski and Wirtgen [24]). Bodlaender, et al. [2] show the following equation (the proof stated in [2] for treewidth applies literally to pathwidth as well).
pwd(Gq) =q(pwd(G) + 1)−1.
(3.1)
Hence, if we apply an absolute approximation algorithm for pathwidth toGq for q large enough, then we can use equation (3.1) to calculate the exact pathwidth ofG.
This idea provides the basis for the next lemma.
Lemma 1. Assume that there is a constantcsuch that|α(G)−pwd(G)| ≤cholds for every cobipartite graph G with minimum degree at least 3. Then the following statements are true.
1. Unless P = NP there is no constant ε, 0 ≤ ε < 1, such that for graphs G withn vertices and minimum degree at least 3,α(G)can be approximated in polynomial-time with an absolute error guarantee ofnε.
2. α-recognitionisNP-hard.
Proof. Part 1. Letεbe a fixed constant with 0≤ε <1. Assume that there exists a polynomial-time algorithmAthat outputs for a given graphGwithnvertices and minimum degree at least 3 an integerA(G) such that|A(G)−α(G)| ≤nε. Thus, we have|A(G)−pwd(G)| ≤nε+c. We choose a constantd≥2 such thatε < d/(d+ 3).
Let Gbe a given cobipartite graph with n≥ 2 vertices wheren is large enough to satisfyn(d+2)ε+c≤n(d+3)ε. Forq=nd+1we form the graphGqusing Construction 1 (observe that sincedis a constant,Gq can be formed in polynomial time). Note that Gq is cobipartite, of minimum degree at leastq−1≥3, and the number of vertices of Gq is exactlynd+2. We apply algorithmAtoGq and get|A(Gq)−α(Gq)| ≤n(d+2)ε. The assumption of the lemma gives|A(Gq)−pwd(Gq)| ≤n(d+2)ε+c≤n(d+3)ε< nd. Using equation (3.1) we get |A(Gq)−q(pwd(G) + 1) + 1| < nd, hence |[(A(Gq) + 1)/q−1]−pwd(G)| < 1/n≤1/2. Hence we can use algorithm A to compute the exact pathwidth ofGin polynomial time. By the aforementioned theorem of Arnborg et al., this is not possible unless P = NP.
Part 2. We reduce pwd-recognition to α-recognition. Let (G, k) be an instance of pwd-recognition. Since pwd-recognitionis NP-hard for cobipartite graphs, we may assume that G is cobipartite. We obtain in polynomial time an instance (G∗, k∗) of α-recognition, puttingG∗=G2c+3 andk∗= (2c+ 3)(k+ 1) + c−1. Observe thatG∗is cobipartite and has minimum degree at least 3. We show that (G, k) is a yes-instance of pwd-recognitionif and only if (G∗, k∗) is a yes-instance of α-recognition; that is, pwd(G) ≤ k if and only if α(G∗) ≤ k∗. First assume pwd(G)≤k. Now by equation (3.1),α(G∗)≤pwd(G∗) +c= (2c+ 3)(pwd(G) + 1)− 1 +c ≤(2c+ 3)(k+ 1) +c−1 = k∗, thusα(G∗) ≤k∗ follows. Conversely, assume α(G∗)≤k∗. We have (2c+ 3)(pwd(G) + 1)−1 = pwd(G∗)≤α(G∗) +c≤k∗+c= (2c+ 3)(k+ 1) +c−1 +c. Hence, pwd(G)≤k+ 2c/(2c+ 3). Since pwd(G) andk are integers, pwd(G)≤kfollows.
We shall use the following two constructions.
Construction 2. Let G denote a graph with n ≥ 2 vertices. We obtain a graphG" from G by replacing each edgeuv of G by three internally disjoint paths u−xi−yi−v, i = 1,2,3, of length 3 (the verticesxi, yi are new vertices); we call such pathsbridges.
Construction 3. Let G denote a graph with n ≥ 2 vertices. We obtain a graphG"" from Gby replacing each edgeuv ofGby a pathu−su,v−v, wheresu,v
is a new vertex.
In Section 5 we show that the following inequalities hold.
pwd(G)≤lin-cwd(G")≤pwd(G) + 4.
(3.2)
In view of Lemma 1 this already implies the NP-hardness of lin-cwd-recognition (see the end of Section 5). However, the gap between cwd(G) and lin-cwd(G) can be arbitrarily large [21, 17]. Hence the hardness result for linear clique-width does not extend directly to clique-width. Fortunately, we can bound the gap between cwd(G") and lin-cwd(G") by a small constant if G" is obtained from a cobipartite graph G of minimum degree at least 2 by means of Construction 2. In particular, for such graphsG" we establish the following inequalities.
cwd(G")≤lin-cwd(G")≤cwd(G") + 18.
(3.3)
We obtain the non-trivial part of inequality (3.3) by means of Construction 3. We show by Lemma 7, Theorem 5, and Lemma 8, respectively, that for every cobipartite graphGof minimum degree at least 2 we have
lin-cwd(G")≤lin-cwd(G"") + 9≤cwd(G"") + 15≤cwd(G") + 18.
(3.4)
The hardest task for showing (3.4) is to bound the linear clique-width ofG""in terms of the clique-width ofG""plus a small constant; this is established in Theorem 5. There we start with an arbitraryk-expression definingG"" and modify thisk-expression in several steps. In each step we introduce only a small number of new labels, and finally we are left with alinear (k+ 6)-expression definingG"".
Consider now the graph parameter
α(G) = cwd(G");
i.e., α(G) is the clique-width of the graph G" obtained from G by Construction 2.
Inequalities (3.2) and (3.3) yield |α(G)−pwd(G)| ≤ 18, hence the assumption of Lemma 1 is met. It is now easy to establish Theorems 1 and 2 as follows.
Proof of Theorem 1. Assume that for a constant ε, 0 ≤ ε < 1, there exists a polynomial-time algorithm A that outputs for a given graph G with n vertices of degree at least 3 an integerA(G) with|A(G)−cwd(G)| ≤nε. For a graphGwithn vertices and minimum degree at least 3,G"has exactlynvertices of degree at least 3;
applying A to G" gives now |A(G")−cwd(G")| =|A(G")−α(G)| ≤ nε. Hence, by the first part of Lemma 1 such algorithmA cannot exist unless P = NP. A similar reasoning applies if the approximation error is bounded by some fixed constant.
Proof of Theorem 2. The second part of Lemma 1 implies thatα-recognitionis NP-hard. We reduceα-recognitionto cwd-recognitionby taking for an instance (G, k) of the former problem the instance (G", k) of the latter problem; obviously α(G) ≤k if and only if cwd(G")≤k. Thus cwd-recognition is NP-hard as well.
The problem is in NP since, given a graphG, we can guess ak-expression and check in polynomial time whether it is indeed ak-expression definingG.
4. Further notation on clique-width. In this section we introduce further (and more technical) notations for cwd-expressions that we will use in the proofs below.
A vertex v of a labeled graphG is a singleton if there is no other vertex of G that has the same label asv. For a cwd-expressiont, we denote byval(t) the labeled
graph defined by t. We denote a cwd-expression which uses at most k labels as a k-expression; for convenience we assume that the k labels are the integers 1, . . . , k.
For a labeled graphGwe denote bylabels(G) the number of labels used inG.
For a cwd-expressiont defining a graphG, we denote by tree(t) the parse tree constructed fromtin the usual way. Fig. 4.1 shows the tree for the cwd-expression for K4given in Section 2.1. The leaves of tree(t) are the vertices ofGwith their initial
ρ2→1 η1,2
⊕ ρ2→1
η1,2
⊕ ρ2→1
η1,2
⊕ 1(v) 2(u)
2(w) 2(x)
Fig. 4.1.Parse tree.
labels, and the internal nodes oftree(t) correspond to the operations oftand can be either binary, corresponding to ⊕, or unary, corresponding to η or ρ. For a node a of tree(t), we denote by tree(t))a* the subtree of tree(t) rooted at a. We denote byt)a*the cwd-expression corresponding totree(t))a*; i.e.,tree(t))a*=tree(t)a*).
Note that in t)a*, and similarly in tree(t)a*), we assume that the operation atais already established (this operation is the leading symbol oft)a*).
For a vertexxofval(t)a*), we say thatxisdead at a, ordead atval(t)a*), if all the edges incident toxinval(t) are included inval(t)a*). Otherwise we say thatxis active ata, oractive at val(t)a*). We say that label&isdead intif it is not involved in anyη-operation int. In other words,&is dead intif there is noη-operation intof the formη","! for any label&".
Let a be a⊕-operation of a cwd-expression t. Ifz is a vertex of val(t)a*) and has label&inval(t)a*) we say that zoccurs at awith label&. Letb andcbe the left and right children ofa, respectively. We say that vertexxoccurs on theleft (right) side of a if it occurs at b (c). We say thata is a 1-⊕-operation if there is exactly one vertex occurring on the left side ofaor there is exactly one vertex occurring on the right side ofa. We say thatais a (>1)-⊕-operation if it is not a 1-⊕-operation.
Thus,tis a linear cwd-expression if all ⊕-operations int are 1-⊕-operations.
LetG"andG""be the graphs obtained fromGby means of Constructions 2 and 3,
respectively. We call the vertices ofG" and G"" which are also vertices of Gregular vertices. We call the vertices ofG"andG""which are not vertices ofGspecial vertices.
5. Comparing pathwidth and linear clique-width. This section is devoted to showing that the pathwidth ofGand the linear clique-width of G" differ at most by 4.
We shall use a characterization of pathwidth by means of graph layouts. A (linear) layout (or arrangement) of a graphG = (V, E) with n vertices is a 1-to-1 map ϕ: V → {1, . . . , n}. For a layout ϕ of G and i ∈ {1, . . . , n} we define four sets of vertices: LG(i, ϕ) is the set of vertices u ∈ V with ϕ(u) ≤ i; we put RG(i, ϕ) = V\LG(i, ϕ). L∗G(i, ϕ) is the set of vertices ofLG(ϕ, i) that have a neighbor inRG(i, ϕ);
symmetrically, RG∗(i, ϕ) is the set of vertices of RG(i, ϕ) that have a neighbor in LG(i, ϕ). Thein-degree and the out-degree of ϕis defined as maxn−1i=1 |R∗G(i, ϕ)|and maxn−1i=1 |L∗G(i, ϕ)|, respectively. Thevertex separation number vsn(G) ofGis defined as the smallest in-degree over all layouts of G(which equals the smallest out-degree over all layouts ofG). It is well-known that pathwidth equals the vertex separation number [25]. Using the notion of vertex separation number, it is easy to see that for every graphGwe have
lin-cwd(G)≤pwd(G) + 2.
(5.1)
Namely, consider a layoutϕofGwith in-degreek. A linear (k+2)-expression defining Gcan be obtained from ϕas follows. We introduce the vertices according to ϕone after the other. At stepi of the construction process we introduce vertexϕ(i) with labelk+1; at this stage all vertices inL∗G(i−1, ϕ) have distinct labels from{1, . . . , k}. We can connect ϕ(i) with its neighbors in L∗G(i−1, ϕ) using separate η-operations.
Thereafter, we change the labels of vertices inL∗G(i−1, ϕ)\L∗G(i, ϕ) to the dead label k+ 2. SinceL∗G(i, ϕ)≤k, at least one labelj ∈ {1, . . . , k}is now available; we change the label of ϕ(i) to j and continue. This process yields a linear (k+ 2)-expression defining G.
For the remainder of this section G denotes a fixed graph with n ≥ 2 vertices
andG" denotes the graph obtained fromGby means of Construction 2.
Lemma 2. If G admits a layout with in-degree k, then the linear clique-width
of G" is at mostk+ 4. Hencelin-cwd(G")≤pwd(G) + 4.
Proof. Assume pwd(G) =k. Hence there exists a layoutϕofGwith in-degreek.
Fori= 1, . . . , n let Γi denote the set of vertices ofG" that belong toLG(i, ϕ) or are of distance at most 2 apart from LG(i, ϕ). Thus, if at least one end of a bridge b belongs toLG(i, ϕ), then both internal vertices ofb belong to Γi. Let ∆i denote the subset of Γi consisting of vertices that are adjacent inG" with vertices outside of Γi. Furthermore, letG"i denote the subgraph ofG" induced by the set Γi.
We inductively obtain linear (k+ 4)-expressionstidefiningG"i,i= 1, . . . , n,such that the labeling ofval(ti) satisfies the following conditions.
1. vertices in Γi\∆i are labeled by 1;
2. vertices in ∆i are labeled by integers from 5. . . , k+ 4;
3. two vertices of ∆i share the same label if and only if both vertices have a common neighbor inG".
Let f : R∗G(1, ϕ) → {5, . . . , k + 4} be an injective map (such map exists since
|RG∗(1, ϕ)| ≤k). The expressiont1 is obtained as follows.
1. We start with the term 2(u) which introduces the vertex u = ϕ−1(1) with label 2.
2. For every pairx, y of vertices that lie on a bridge between uand some vertex v∈RG∗(1, ϕ) we add the following sequence of operations.
2.1. A 1-⊕-operation that adds vertexxwith label 3.
2.2. Anη2,3-operation that connectsuandx.
2.3. A 1-⊕-operation that addsy with label 4.
2.4. Anη3,4-operation that connectsxandy.
2.5. Aρ3→1-operation that givesxthe dead label 1.
2.6. Aρ4→f(v)operation that givesy the labelf(v).
3. Finally, we add aρ2→1-operation that givesuthe dead label 1.
The linear (k+4)-expressiont1definesG"1and the claimed properties are evidently satisfied. Now assume that we have already a (k+ 4)-expressionti−1 defining G"i−1 for somei∈ {2, . . . , n}such thatval(ti−1) satisfies the claimed properties.
For vertices v ∈ R∗G(i, ϕ) let ∆i−1(v) denote the set of vertices in ∆i−1 that are adjacent tov in G". Let u=ϕ−1(i). By assumption, there is an injective map
f" : R∗G(i−1, ϕ) → {5, . . . , k+ 4} such that all vertices in ∆i−1(u) have the same
labelf"(u) inval(ti−1), and no other vertex ofval(ti−1) is labeled withf"(u). Since
|RG∗(i, ϕ)| ≤ k, we can define an injective map f : R∗G(i, ϕ) → {5, . . . , k+ 4} with f(v) =f(v)" forv∈R∗G(i−1, ϕ)∩R∗G(i, ϕ).
We extendti−1 to a (k+ 4)-expressionti definingG"i, adding the following oper- ations immediately above the root oftree(ti−1).
1. Add a 1-⊕-operation that introduces the vertexuwith label 2 (uis now the only vertex labeled with 2).
2. Add anηf!(u),2-operation that connects all vertices in ∆i−1(u) withu.
3. Add aρf!(u)→1-operation that gives all vertices in ∆i−1(u) the dead label 1.
4. As above, for every pairx, y of vertices that lie on a bridge between uand somev∈R∗G(1, ϕ) we add the sequence of operations 2.1–2.6 given above.
5. Finally, we add aρ2→1-operation that givesuthe dead label 1.
It is straightforward to verify that the obtained linear (k+ 4)-expression defines
G"i and that the labeling ofval(ti) satisfies the claimed properties.
SinceG"n=G", it follows that the linear clique-width ofG" is at mostk+ 4.
The next lemma will allow us to bound the pathwidth ofGin terms of the linear clique-width of G", a result inverse to Lemma 2. To this end let us fix a linear k-expression t defining G". Since t is linear it gives raise to a sequence t1, . . . , ts
of linear k-expressions such that t1 defines an initial k-graph, ts =t, and for each i∈ {2, . . . , s},ti is obtained fromti−1 by adding a ρ-operation, a η-operation, or 1-
⊕-operation that introduces a new vertex. For every edgeeofG" letj(e) := min{1≤ j≤s:e∈val(tj)}. We call a bridgeu−x−y−v ofG" well-behaved (relative tot) ifuis a singleton inval(tj(ux)) andv is a singleton inval(tj(yv)).
Lemma 3. At least one of any three parallel bridges ofG" is well-behaved.
Proof. For an edgeuv of Glet b1, b2, b3 denote the three corresponding bridges
of G", where bi is the bridge u−xi−yi −v, i = 1,2,3. For i = 1,2,3 we put
αi= max(j(uxi), j(yiv)).
Claim A: j(uxi)and j(yiv) must be distinct for i= 1,2,3. Otherwise, either u would have the same label asyior the same label asvinval(tj(uxi)). In the first case, the addition of the edgeyiv causes the addition of the edgeuv. In the second case, the addition of the edgeyivcauses the addition of the edgeyiu. However, neitheruv noryiuis present inG". Hence Claim A is shown.
Claim B: if j(uxi) < j(yiv), then u is singleton in val(tj(uxi)) for i = 1,2,3.
Assume to the contrary that there is a vertexw%=uin val(tj(uxi)) which shares the label withu. It follows thatG" contains the edgewxi, hencew=yi. This, however, implies that val(tj(yiv)) contains the edge uv, a contradiction. Hence Claim B is shown.
Now we proceed with the proof of the lemma. We consider two cases.
Case 1: |{α1, α2, α3}| ≤ 2. We assume, w.l.o.g., α1 = α2 = j(y1v). Clearly α2 = j(y2v), since otherwise, if α2 = j(ux2), then some of the edges uy1, uv were
present inG". Letwbe a vertex ofval(tj(y1v)) that shares the label withv. It follows
that G" contains the edges wy1 and wy2; hence w = v. Thus v is a singleton in
val(tj(y1v)). Since j(ux1)< j(y1v), it follows from Claim B thatu is a singleton in val(tj(ux1)). Hence the bridgeb1is well-behaved.
Case 2: |{α1, α2, α3}|= 3. We assume, w.l.o.g., thatj(y1v) =α1< α2< α3. Subcase 2a: j(y2v) > j(y1v) or j(y3, v) > j(y1v). W.l.o.g., j(y2v) > j(y1v).
Similarly as above we conclude that for any vertexw ofval(tj(y1v)) that shares the label with v, the edgesy1w and y2w are present in val(tj(y1v)) and val(tj(y2v)), re- spectively. Hence w=v, and so v is a singleton in val(tj(y1v)). Furthermore, since j(ux1)< j(y1v), it follows by Claim B thatuis a singleton inval(tj(ux1)). Hence the bridgeb1 is well-behaved.
Subcase 2b: j(y2v) ≤j(y1v) and j(y3, v)≤j(y1v). It follows that α2 =j(ux2) andα3=j(ux3). We show thatuis a singleton inval(tj(ux2)). Letwbe a vertex of val(tj(ux2)) that shares the label with u. Consequently, the edges wx2 and wx3 are present inval(tj(ux2)) and val(tj(ux3)), respectively. Thus u=w and souis indeed a singleton in val(tj(ux2)). Using a symmetrical version of Claim B, we conclude from j(y2v) < j(ux2) that v is a singleton in val(tj(y2v)). Hence the bridge b2 is well-behaved.
Lemma 4. If G" has linear clique-widthk, then Gadmits a layout of in-degree at mostk. Hencepwd(G)≤lin-cwd(G").
Proof. Letkbe the number of labels used int. For a vertexvofGletβ(v) denote the smallest integer in {1, . . . , s} such that v is not a singleton of val(tβ(v)). Note that β(v) is defined for every vertexv of G, since we assume thatGhas more than one vertex and all vertices ofval(t) have label 1. Note also that ifβ(v) =β(v") =j holds for two verticesv, v" ofG, thenv andv" have the same label inval(tj), but no other vertex inval(tj) shares its label withvandv" (eithervandv" are singletons in val(tj−1) and one of the two vertices is relabeled with the other’s label inval(tj), or one of the two vertices is a singleton inval(tj−1) and the other vertex is introduced in val(tj) with the same label). Letϕbe a layout ofGsatisfyingϕ(v)< ϕ(v") whenever β(v)< β(v").
We show that the in-degree of the layoutϕis at mostk. Choosei∈ {1, . . . , n−1} arbitrarily. We show that |R∗G(i, ϕ)| ≤k. Let w= ϕ−1(i), j =β(w), and consider the k-graph val(tj). By construction, the vertices in LG(i, ϕ) are not singletons of val(tj). We assign to every vertex v ∈R∗G(i, ϕ) a labelf(v)∈ {1, . . . , k} as follows (it will turn out thatf is an injective map). Choose arbitrarily a vertexv∈R∗G(i, ϕ).
By definition,vis inGadjacent to a vertexu∈LG(i, ϕ). Thusuandvare joined by three parallel bridges inG". By Lemma 3, at least one of the bridges betweenuand v, sayb= (u, xv, yv, v), is well-behaved int. For verticesz ofval(tj) let&(z) denote the label ofzin val(tj). We put
f(v) =
&(v) ifv∈val(tj); (c1)
&(yv) ifv /∈val(tj) andyv ∈val(tj); (c2)
&(xv) ifv, yv∈/val(tj). (c3)
Sinceuis not a singleton inval(tj), the edgeuxv must already be present inval(tj) as the bridgeu−xv−yv−vis well-behaved. Consequently the above case distinction is exhaustive. We split the setR∗G(i, ϕ) into setsC1,C2, andC3, such that a vertexv belongs toCi iff(v) is assigned by means of the above case (ci). We further splitC1
into setsC1= and C1< such thatv ∈C1 belongs to C1= ifβ(w) =β(v) andv belongs toC1< ifβ(w)< β(v).
To show that f is an injective map, suppose to the contrary that f(v) = f(v") for two distinct vertices v, v" ∈R∗G(i, ϕ). Since the vertices of C1< are singletons in val(tj), v, v" ∈/ C1< follows. For anyv ∈ C3, the vertexxv is a singleton in val(tj) since the edgexvyv is still missing, hence v, v" ∈/ C3. Furthermore, v and v" cannot both belong to C1= since then both would share the label with w in val(tj), but as seen above, any v ∈C1= shares its label only withw. If v ∈C1= and v" ∈C2, then when the edgeyv!v" is established, also the edge vv" is established, a contradiction, sincevv" is not an edge ofG". Hence we are left with the casev, v" ∈C2. Thusf(v) is the label ofyvandf(v") is the label ofyv!. The edgesyvv, yv!v"are not yet present in val(tj) since the vertices v, v" are not yet present inval(tj) either. If at a further step the edgeyvv is added, also the edgeyvv" is added, but G" does not contain the
edgeyvv", a contradiction. Thusf :R∗G(i, ϕ)→ {1, . . . , k}is indeed an injective map,
and so|R∗G(i, ϕ)| ≤kfollows. Hence we have shown that pwd(G) = vsn(G)≤k.
Having established inequality (3.2), we can use a similar reasoning as outlined in Section 3 (however, settingα(G) = lin-cwd(G")) to establish the following results.
Theorem 3. Letεbe a constant with0≤ε <1. The linear clique-width of graphs withn vertices of degree greater than 2 cannot be approximated by a polynomial-time algorithm with an absolute error guarantee ofnε unlessP = NP.
In particular, there is no polynomial-time absolute approximation algorithm for linear clique-width unlessP = NP.
Theorem 4. lin-cwd-recognitionisNP-complete.
6. Comparing Construction 2 and Construction 3. For this section letG denote a graph with minimum degree at least 2. We show that the clique-width of
G""is bounded by the clique-width ofG" plus a small constant, and that the converse
is true for linear clique-width.
6.1. FromG"" to G".
Property 1. Lett be a linear cwd-expression defining G"". We say that t has Property 1if the following two conditions are satisfied.
Condition 1.1: Everyη-operationaintestablishes some edge which is not estab- lished by anotherη-operation aboveain tree(t).
Condition 1.2: For every two regular verticesxandythere is no nodeaintree(t) such thatxandy are active ataand have the same label ata.
Lemma 5. Let t be a lineark-expression definingG"". Then there exists a linear (k+ 2)-expression definingG"" which has Property 1.
Proof. Lett be a lineark-expression definingG"". Since we can remove η-opera- tions that invalidate Condition 1.1 fromt, we may assume that allη-operations int satisfy this condition. Letx and y be two regular vertices such that there exists a nodeaintsuch thatxandyhave the same label ataand are active ata. Letb the lowest node intree(t) corresponding to an operation which unifies the labels ofxand y. Clearlyb corresponds to either aρor a 1-⊕-operation. Supposeb corresponds to a 1-⊕-operation. This operation introduces either xor y (say that it introducesx).
Since x and y have the same label at b it follows that each neighbor of x is also a neighbor ofy. However, since Ghas minimum degree at least 2, there is a neighbor
ofxinG""which is not a neighbor of y, a contradiction.
Let b1 be the child of b in tree(t). Clearlyx and y are active at b. Since sx,y
is the unique vertex in G"" which is adjacent to both x and y, it follows that if we add the edges connectingxandy tosx,yimmediately aboveb1, thenxandywill not be active atb. We show below how to construct an expressiont1 which achieves this goal.
Lett"1be the expression obtained by removingsx,yfromt. Lett1be the expression obtained from t"1 by adding immediately above b1 the vertex sx,y with label k+ 2, then adding twoη-operations which connectsx,yto bothxandyand then renaming the label ofsx,yto k+ 1. (Note thatk+ 1 will be a dead label, i.e., no edges will be added to a vertex having labelk+ 1.) Since both edges connecting sx,y to xand y already exists atval(t1)b*), it follows thatxandy are not active atval(t1)b*).
Repeating the above construction for every pair of regular verticesxandy which have the same label at a nodeaoftree(t) and are active ata, we finally get a linear (k+ 2)-expressiont" which definesG""and satisfies Property 1.
Note that whenever vertex sx,y gets labelk+ 2 at node a of t" it is the unique vertex having this label inval(t")a*) and thus, it is possible to connect it toxand y using twoη-operations.
Lemma 6. Let t be a lineark-expression definingG"" that has Property 1. Then there exists a linear(k+ 7)-expression definingG".
Proof. Lettbe a lineark-expression definingG""that has Property 1. Lets=sx,y
be a special vertex ofG"". Thus the neighbors of sin G"" are the regular verticesx and y. Lete1 ande2 denote the edges connectings to xand y, respectively. If the edgese1ande2are established intby the sameη-operation, then there is a nodeain tsuch that bothxandyhave the same label ataand are active ata, a contradiction.
Thus, we can assume without loss of generality that the edgee1is established before e2 in t. Let a denote the lowest node in tree(t) corresponding to the η-operation which establishes the edge e1 in t. Since x and s must have unique labels at a, it follows from Property 1 that node a is the only η-operation in t which connects x to s. Lett"1 denote the expression obtained by removings fromt. Lett1denote the expression obtained fromt"1 by replacing the node a with the following sequence of operations:
1. Add verticess1, . . . , s6 with labelsk+ 2, . . . , k+ 7, respectively.
2. Addη-operations connectings1,s2, ands3 tox.
3. Addη-operations connectings1tos4,s2 tos5, ands3 tos6.
4. Addρ-operations which rename the labels ofs1, s2, ands3 to k+ 1 (k+ 1 is used as a dead label).
5. Addρ-operations which rename the labels ofs4,s5. ands6to&, where&is the label thatshas in val(t)a*).
It is easy to check thatt1 defines the graph obtained from G"" by replacing the path of length twox−s−ywith the 3 paths of length 3,x−si−si+3−y,i= 1,2,3.
Repeating the above construction for every special vertex s of G"", we finally obtain a linear (k+ 7)-expressiont" which definesG".
Note that whenever verticess1, . . . , s6 get labelsk+ 2, . . . , k+ 7 at nodea oft"
they are the unique vertices having these labels in val(t")a*) and thus, it is possible to establish all the connections and renaming mentioned in steps 2–5 above.
This completes the proof of the lemma.
Lemma 7. lin-cwd(G")≤lin-cwd(G"") + 9.
Proof. Suppose lin-cwd(G"") =k, i.e., there exists a linear k-expression t which definesG"". By Lemma 5 there exists a linear (k+ 2)-expressiont1which definesG""
and has Property 1. By Lemma 6 there exists a linear (k+ 9)-expression t2 which definesG". Thus lin-cwd(G")≤k+ 9.
6.2. FromG" to G"".
Lemma 8. cwd(G"")≤cwd(G") + 3.
For proving this lemma we shall use the following definitions and lemma.
LetGbe a graph and let D(G) denote the set of graphs which can be obtained fromGby replacing each edge ofGeither with a path of length two or with a path of length three. Clearly, the graphG""belongs toD(G) and is obtained by replacing all edges ofGwith a path of length two. For each graphG∗ inD(G) we call the vertices of G∗ which are also vertices of G regular vertices and we call the other vertices of G∗ special vertices.
Property 2. Lettbe ak-expression defining a graphG∗inD(G). We say thatt hasProperty 2if the following conditions hold:
Condition 2.1: There is no η-operation in t which uses label 1, i.e, there is no η1,"-operation intfor any label &. In other words, 1 is a dead label.
Condition 2.2: If label 2 is used int, then it is used as follows: a special vertex (says) is introduced with label 2 using a 1-⊕-operation saya, such thatsis the only vertex having label 2 at a. Above a in tree(t) there is a sequence of one or more η-operations followed by aρ2→"-operation where&is any label different from 2 and 3.
Condition 2.3: If label 3 is used in t then it is used as follows: a regular vertex (say r) is introduced with label 3 using a 1-⊕-operation, say a, such that r is the only vertex having label 3 ata. Above aintree(t) there is a sequence of operations which can be either η, ρ, or 1-⊕-operations introducing special vertices, followed by aρ3→"-operation where&is any label different from 2 and 3.
Condition 2.4: No regular vertex ever gets label 2 and no special vertex ever gets label 3.
Observation 1. Let G∗ be a graph in D(G) and let cwd(G∗) = k. Then there is a(k+ 3)-expressiont" defining G∗ which has Property 2.
Proof. Lettbe ak-expression definingG∗. Lett" be thek+3-expression obtained fromtby replacing all occurrences of the labels 1,2 and 3 with the labelsk+ 1,k+ 2 andk+ 3, respectively. Clearlyt" definesG∗. Since the labels 1,2 and 3 are not used
int", it is obvious thatt" has Property 2.
The following is the key lemma for proving Lemma 8.
Lemma 9. Let G∗ be a graph inD(G)and let t be a k-expression which defines G∗ and has Property 2. Let a be a lowest node in tree(t) such that there exists an induced pathx−p−q−y in G∗ (x, y are regular vertices) andx, p, q, y occur ata.
Then there exists a k-expression t1 which has Property 2 and defines the graph G∗1 obtained fromG∗ by replacing the pathx−p−q−y with a path x−s−y wheresis a new special vertex.
Proof. Let a and x, p, q, y as in the statement of the lemma. In each of the following cases we obtain ak-expressiont1which defines G∗1 and has Property 2. In all cases it is easy to see that the expressiont1 obtained has Property 2.
Case 1: supposexandy occur on different sides ofa. Assume w.l.o.g. thatxis on the left side ofaand yis on the right side ofa.
Case 1.1: suppose thatpandqoccur on the same side ofa. Assume without loss of generality that bothpandq occur on the left side ofa. Leta1 denote the lowest node intree(t) such that bothxandpare int)a1*. Leta2 denote the lowest node in tree(t) such that bothxandq are int)a2*. By the above assumptions both a1and a2 are descendants ofain tree(t).
Case 1.1.1: suppose a1is a proper descendant of a2 intree(t). Ifxandq have the same label ata2 it follows thaty must be int)a2*, a contradiction. Thuspand q must have unique labels at a2. Let&p and &q denote the labels of pand q at a2, respectively.
Case 1.1.1.1: suppose x has a unique label (say &x) at a2. In this case, t1 is
obtained fromtas follows:
1. Add the following sequence operations immediately abovea2: 1.1. Anη"x,"p-operation which connectsxtop.
1.2. Aρ"p→"q-operation which renames the label ofpto the label ofq.
2. Omitq.
Case 1.1.1.2: Suppose xdoes not have unique label ata2. Thus the edge con- necting x to p already exists at val(t)a2*). In this case, t1 is obtained from t as follows:
1. Add immediately abovea2 aρ"p→"q-operation which renames the label ofpto the label ofq.
2. Omitq.
In both cases 1.1.1.1 and 1.1.1.2,pis connected toy since after pgets the label ofq, the η-operation aboveawhich connectsqto y will connectptoy. Thus,pcan be considered as the new special vertexsin G∗1 and the expressiont1 definesG∗1.
Case 1.1.2: Suppose a1 is equal to a2. In this casex and q must have unique labels ata2;t1 is obtained fromtas follows:
1. Add immediately abovea2 anη"x,"q-operation which connectsxandq.
2. Omitp.
In this case it is easy to see thatt1 definesG∗1 andqis the new special vertexs.
Case 1.1.3: supposea2 is a proper descendant ofa1 intree(t). Sinceyis not in t)a1*, x, p, and qmust have unique labels ata1. Let&x,&p, and&q denote the labels ofx,pandqat a1, respectively. In this case,t1 is obtained fromtas follows:
1. Add the following sequence operations immediately abovea1: 1.1. Anη"x,"p-operation which connectsxtop.
1.2. Aρ"p→"q-operation which renames the label ofpto the label ofq.
2. Omit q.
As in the previous cases it is easy to see thatt1definesG∗1andpis the new special vertexs.
Case 1.2: suppose thatpandqoccur on different sides ofa.
Case 1.2.1: suppose poccurs on the left side ofaand qoccurs on the right side of a. It is easy to see that at least one of pand q must have a unique label at a.
Assume w.l.o.g. that q has a unique label (say &q) at a. Let &p and &y denote the labels thatpand y have ata, respectively. Note thaty is the only vertex which can have the same label aspata. In this case,t1is obtained fromtas follows:
1. Make changes tot such that y will have label &q at a. In particular letc be the lowest ⊕-operation in tree(t) which contains both y and q. Add a ρ-operation immediately abovecwhich renames the label ofyatcto the label ofqatc. Note that sinceq has a unique label&q ata, we may assume that the label ofq atc is also &q. Then follow the path from c to a in tree(t) and for each node d corresponding to anη"1,"2-operation such thaty has label&1atd, add anη"q,"2-operation immediately aboved. Thus, after this step y is connected to all the vertices (except q) which it was connected inval(t)a*) and has label&q at a.
2. Omitq.
3. After the above changes toy, the label&pofpatais unique. Add the following sequence of operations immediately above a:
3.1. Anη"p,"q-operation which connects yto p.
3.2. Aρ"q→"y-operation which renamesy to the label it has inval(t)a*).
By steps 1 and 3.2 above it is clear that all the vertices (except q) which are connected toyintare also connected toy int1. Thus,t1definesG∗1andpis the new
special vertexs.
Case 1.2.2: suppose poccurs on the right side ofaand qoccurs on the left side of a. Sincepis adjacent just toxand q, it follows that eitherxand q have unique labels at a or have the same label at a. If xand q have the same label at a, then there is no way to connecty toqwithout connecting it also tox, a contradiction. We conclude that the labels ataofp,q,x, and y(say&p, &q,&x and&y, respectively) are unique. In this caset1is obtained fromtby omittingqand adding anη"p,"y-operation immediately abovea.
Case 2: suppose x and y occur on the same side of a. Assume without loss of generality thatxandy occur on the left side ofa.
Case 2.1: supposepandqoccur on the same side ofa. Sinceais the lowest node in tree(t) which contains x, y, p, and q, it follows thatp and q must occur on the right side ofa. As in case 1.2.2 it is easy to see that the labels ataof p, q,xand y (say&p,&q, &x, and &y) are unique. In this case t1 is obtained from t by omittingq and adding anη"p,"y-operation immediately abovea.
Case 2.2: suppose pandqoccur on different sides of a. Assume without loss of generality thatpoccurs on the left side ofaandq occurs on the right side ofa. Let a1denote the lowest node intree(t) which contains bothxandp. Leta2 denote the lowest node intree(t) which containsxandy.
Case 2.2.1: suppose a1 is equal to a2 or a2 is a proper descendant of a1. In this case it is easy to see that x, y and pmust have unique labels ata1 (say&x, &y, and&p, respectively). In this caset1is obtained from tby omittingqand adding an η"p,"y-operation immediately abovea1.
Case 2.2.2: supposea1 is a proper descendant ofa2.
Case 2.2.2.1: supposeyhas unique label ata2 (say&y). In this casepmust have unique label at a2 (say &p) and t1 is obtained from t by omitting q and adding an η"p,"y-operation immediately abovea2.
Case 2.2.2.2: supposey does not have unique label ata2. Let&p and &y denote the labels ofpandy ata2, respectively. Sinceqis adjacent just toy andp, it follows thatpis the only vertex which can share the label ofy ata2. Thus,&p=&y. Assume without loss of generality thatyis on the right side ofa2 andxandpare on the left side of a2. Letb2 denote the right child ofa2 in tree(t). Note that the complicated handling of this case (as described below) is needed when xis active at a2 and has the same label as another vertex which is on the right side ofa2. Sinceq is the only vertex which is adjacent toy andp, it follows that all the vertices which are adjacent to y (exceptq) must be inval(t)b2*). Let U denote the set of all vertices (except q) which are adjacent toy. Since y is regular vertex, all vertices in U must be special and have degree exactly 2. For each vertexuinU, letother(u) denote the neighbor ofuwhich is noty. LetU1 denote the set of all verticesuinU such thatother(u) is inval(t)b2*) and letU2=U\U1. LetU11 denote the set of all verticesuin U1 such that the lowest node in tree(t) which contains uand other(u) does not containy.
LetU12=U1\U11.
In this caset1is obtained fromt as follows:
1. Omitqand all vertices ofU2.
2. Letcdenote the lowest node intree(t) which containsy. Follow the path from ctob2intree(t) and omit anyη"1,"2-operation such that the label ofy at that point is&1.
3. Repeat the following step for each uin U11: let c denote the lowest node in tree(t) which containsuandother(u). Letddenote the lowest node intree(t) which
containsyandu. Sinceuis inU11,cis a descendant ofd. Thus,uandother(u) have unique labels atc (say &u and&, respectively). Add an η"u,"-operation immediately above c which connects u and other(u). Add a ρ-operation immediately above d which renames the label ofuto the label ofy atd. Thus, after step 3 each vertexu inU11is connected toother(u) and has label&y at a2.
4. Repeat the following step for each uin U12: let c denote the lowest node in tree(t) which containsuandother(u).
4.1. Suppose other(u) is a special vertex. If other(u) does not have a unique label at c then its label at c must be equal to the label of y at c, a contradiction, sinceq distinguishes y and other(u). Thus, other(u) must have unique label at c.
If udoes not have unique label at c, then the label of uat c must be equal to the label of the unique regular vertex (say z) which is adjacent to other(u). But then vertices of the induced path z−other(u)−u−y of G∗ occur at a2, and since a2
is a descendant of a, we have a contradiction to the selection ofa as a lowest such node with that property. We conclude that bothuandother(u) have unique labels at c. Thus, in this case add anη-operation immediately above c connecting uand other(u) and above it add a ρ-operation which renames the label ofu to the label thaty has at that point.
4.2. Supposeother(u) is a regular vertex. Sincethas Property 2, it follows that either label 2 is not used atcor uis the only vertex having label 2 atc. In this case omitufrom tand add the following sequence of operations immediately abovec:
4.2.1. A 1-⊕-operation introducinguwith label 2.
4.2.2. Anη2,"-operation, where&is the unique label thatother(u) has atc.
4.2.3. Aρ2→"!-operation where&" is the unique label thaty has atc.
Thus, after step 4 each vertexuinU12is connected toother(u) and has label&y
ata2.
5. Omit y from t and add the following sequence of operations immediately abovea2:
5.1. A 1-⊕-operation which introduces y with label 3. Note that since t has Property 2 label 3 is not used ata2.
5.2. Anη3,"y-operation connectingy topand all the vertices inU1.
5.3. Aρ"y→1-operation renamingpand all the vertices inU1 to a dead label.
5.4. For each vertexuin U2add the following sequence of operations:
5.4.1. A 1-⊕-operation introducinguwith label 2.
5.4.2. Anη2,3-operation connectinguandy.
5.4.3. Aρ2→"-operation where&is the label that uhas int ata2.
Thus after step 5.4 all the vertices inU2 are connected toy and have the same label as they have intat a2.
5.5. Aρ3→1-operation renaming the label ofy to a dead label.
Each vertexuinU1is connected toother(u) in step 3 or in step 4 and is connected toyin step 5.2. Each vertexuinU2is connected toyat step 5.4.2 and theη-operation intabovea2which connectsutoother(u) also exists int1and connectsutoother(u) since after step 5.4 the label ofuis the same as its label ata2in t.
Thus,t1 definesG∗1 andpis the new special vertexs.
This completes the proof of Lemma 9.
Proof of Lemma 8. Suppose cwd(G") =k. LetG"1 denote the induced subgraph
ofG"obtained by removing fromG"for every edgee=xyofG, the two pairs of vertices
pi, qi,i= 1,2, wherex−pi−qi−y are two of the three paths of length 3 betweenx andy. Since G"1 is an induced subgraph ofG", it follows that cwd(G"1)≤k. Clearly,
