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Splitting Cubic Circle Graphs Discussiones Mathematicae Graph Theory 36 (2016) 723–741 doi:10.7151/dmgt.1894 SPLITTING CUBIC CIRCLE GRAPHS Lorenzo Traldi Lafayette College Easton, Pennsylvania 18042 e-mail: [email protected] Abstract We show that every 3-regular circle graph has at least two pairs of twin vertices; consequently no such graph is prime with respect to the split de- composition. We also deduce that up to isomorphism, K4 and K3,3 are the only 3-connected, 3-regular circle graphs. Keywords: circle graph, split decomposition, regular graph. 2010 Mathematics Subject Classification: 05C62. 1. Introduction Circle graphs have been introduced several times, in several contexts. The inter- section graph of a family of chords in a circle seems to have first been mentioned in print by Zelinka [19]; he gave credit for the idea to Kotzig, whose seminal work [15] founded the special theory of 4-regular graphs. Brahana’s separation matrix [5] — in essence, the adjacency matrix of a circle graph — was intro- duced decades earlier, in connection with the geometry of surfaces. Circle graphs achieved broad recognition in the 1970s, when Even and Itai [10] considered circle graphs in relation to the analysis of permutations using stack and queues; Bouchet [2] and Read and Rosenstiehl [17] discussed the interlacement graphs of double occurrence words in connection with the famous Gauss problem of characterizing generic curves in the plane; Cohn and Lempel [6] related the cycle structure of a certain kind of permutation to the GF (2)-nullity of an associated link relation matrix (which is also the adjacency matrix of a circle graph). An account of the early combinatorial theory appears in Golumbic’s classic book [14]. Definition 1. Let W = w1 ··· w2n be a double occurrence word, i.e., a sequence in which each of n letters appears twice. Then the interlacement graph I(W ) is a graph with n vertices, labeled by the letters appearing in W . Two vertices a 724 L. Traldi and b of I(W ) are adjacent if and only if the corresponding letters appear in W in the order abab or baba. A simple graph that can be realized as an interlacement graph of a double occurrence word is a circle graph. During the last forty years the theory of circle graphs has been sharpened considerably. Polynomial-time recognition algorithms were developed before the new millennium by Bouchet [3], Naji [16] and Spinrad [18]. More recently, Cour- celle [7] has observed that circle graphs are well described in the framework of monadic second-order logic, and Gioan, Paul, Tedder and Corneil have provided the first subquadratic recognition algorithm [12, 13]. Special properties of certain types of circle graphs have been investigated by Bonomo, Dur´an, Grippo and Safe [1] and Daligault, Gon¸calves and Rao [9]. The crucial tool used to design recognition algorithms for circle graphs is the split decomposition of Cunningham [8]. We recall only the basic definition here, and refer to the literature ([7, 8, 13] for instance) for thorough explanations of this important concept. Definition 2. Let G be a simple graph. A split (V1, X1; V2, X2) of G is given by a partition V (G) = V1 ∪ V2 with |V1|, |V2| ≥ 2 and subsets Xi ⊆ Vi with these properties: the complete bipartite graph with vertex-classes X1 and X2 is a subgraph of G, and G does not have any other edge from V1 to V2. Every simple graph of order 4 has a split. A graph with five or more vertices that has no split is said to be prime. Three types of splits are particularly simple. Let G be a graph with four or more vertices. • Suppose v1 and v2 are twin vertices of G, i.e., they have the same neighbors outside {v1,v2}. Then G has a split with V1 = X1 = {v1,v2}. • Suppose G is not connected. Let H be a union of some but not all connected components of G, such that H includes at least two vertices. If h ∈ V (H) then either G has a split with V1 = V (H) and X1 = ∅ or G has a split with V1 = V (H − h) and X2 = {h}. • Suppose G has a cutpoint v. Let H be a union of some but not all connected components of G − v, such that H includes at least two vertices. Then G has a split with V1 = V (H) and X2 = {v}. A fundamental part of the theory of circle graphs and split decompositions is the following operation, which is motivated by the properties of double occurrence words [2, 15, 17]. We use N(v) to denote the open neighborhood of v in G, i.e., the set of vertices w = v such that vw ∈ E(G). Definition 3. Let v be a vertex of a simple graph G. Then the local complement of G with respect to v is the graph Gv with V (Gv) = V (G) and E(Gv) = {xy | either x∈ / N(v) and xy ∈ E(G) or x,y ∈ N(v) and xy∈ / E(G)}. Splitting Cubic Circle Graphs 725 In some references this operation is called simple local complementation, to distinguish it from a related operation that involves looped vertices. We consider only simple graphs in this paper, so we need not be so careful here. Two important properties of local complementation are that G is a circle graph if and only if Gv v is a circle graph, and that G has a split (V1, X1; V2, X2) if and only if G has a split (V1, Y1; V2, Y2). A central result involving circle graphs is Bouchet’s characterization by means of obstructions [4]. Recall that local equivalence is the equivalence relation gen- erated by isomorphisms and local complementations. Theorem 4 [4]. A simple graph G is a circle graph if and only if no graph locally equivalent to G has one of the graphs of Figure 1 as an induced subgraph. Figure 1. Bouchet’s obstructions: W5, BW3 and W7. Observe that W5 and W7 are locally equivalent to the graphs pictured in Figure 2, both of which are prime, 3-connected, and 3-regular. The purpose of this paper is to prove that these local equivalences are no mere coincidence. Theorem 5. Let G be a 3-regular circle graph. Then G has at least two disjoint pairs of twin vertices. Figure 2. Prime, 3-connected cubic graphs locally equivalent to W5 and W7. Theorem 5 immediately implies the following. Corollary 6. Let G be a 3-regular circle graph. Then G is not prime. In Section 4 we deduce another consequence of Theorem 5. Corollary 7. Let G be a 3-regular circle graph, which is not isomorphic to K4 or K3,3. Then G is not 3-connected. 726 L. Traldi 2. Three Lemmas In this section we recall three elementary results about double occurrence words and circle graphs. The reader who has not seen them before will have no trouble devising proofs. Lemma 8. If G is a circle graph with a vertex v, then G − v and Gv are also circle graphs. Lemma 9. If W ′ is obtained from a double occurrence word W by some sequence of cyclic permutations w1 ··· w2n → wiwi+1 ··· w2nw1 ··· wi−1 and reversals w1 ··· w2n → w2nw2n−1 ··· w2w1, then I(W )= I(W ′). Double occurrence words related as in Lemma 9 are said to be cyclically equivalent. By the way, the converse of Lemma 9 is false in general; a complete characterization of double occurrence words with the same connected interlace- ment graph has been provided by Ghier [11]. Lemma 10. Let G1 and G2 be disjoint simple graphs, and let G be a graph obtained by attaching G1 to G2 with a single edge. Then G is a circle graph if and only if both G1 and G2 are circle graphs. 3. Proof of Theorem 5 Before beginning the proof, we might mention that we use standard terms of graph theory — adjacency, component, cycle, path, etc. — without bothering to state their definitions. We should also mention that we sometimes say “two pairs” rather than “two disjoint pairs” while discussing Theorem 5. In fact, the theorem is equivalent to the weaker-seeming assertion that every cubic circle graph has two pairs of twin vertices. For if a cubic graph has two intersecting pairs of adjacent twins, then the three twin vertices and their shared neighbor constitute a 4-clique, which must be a whole connected component; the 4-clique provides three disjoint pairs of twin vertices. If a cubic graph has two intersecting pairs of nonadjacent twins, then the three twin vertices and their three neighbors constitute a 3,3-biclique, which must again be a whole connected component; the biclique provides nine disjoint pairs of twin vertices. (An easy argument shows that a pair of adjacent twins cannot intersect a pair of nonadjacent twins in any graph.) Splitting Cubic Circle Graphs 727 3.1. A minimal counterexample is 2-connected Let G be a minimal counterexample to Theorem 5, i.e., a cubic circle graph that does not have two disjoint pairs of twin vertices, with the smallest possible number of vertices. Then G must certainly be connected, for if not then each connected component of G is itself a smaller counterexample. Proposition 11. Let G be a cubic circle graph that does not have two pairs of twin vertices, and is of the smallest order for such a graph. If G is not 2- connected, then G has precisely two cutpoints, which are adjacent. z x y z y Figure 3.
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