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Generalizing Cographs to 2-Cographs

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Generalizing Cographs to 2-Cographs GENERALIZING COGRAPHS TO 2-COGRAPHS JAMES OXLEY AND JAGDEEP SINGH Abstract. In this paper, we consider 2-cographs, a natural general- ization of the well-known class of cographs. We show that, as with cographs, 2-cographs can be recursively defined. However, unlike cographs, 2-cographs are closed under induced minors. We consider the class of non-2-cographs for which every proper induced minor is a 2-cograph and show that this class is infinite. Our main result finds the finitely many members of this class whose complements are also induced-minor-minimal non-2-cographs. 1. Introduction In this paper, we only consider simple graphs. Except where indicated otherwise, our notation and terminology will follow [5]. A cograph is a graph in which every connected induced subgraph has a disconnected com- plement. By convention, the graph K1 is assumed to be a cograph. Re- placing connectedness by 2-connectedness, we define a graph G to be a 2-cograph if G has no induced subgraph H such that both H and its com- plement, H, are 2-connected. Note that K1 is a 2-cograph. Cographs have been extensively studied over the last fifty years (see, for example, [6, 12, 4]). They are also called P4-free graphs due to following characterization [3]. Theorem 1.1. A graph G is a cograph if and only if G does not contain the path P4 on four vertices as an induced subgraph. In other words, P4 is the unique non-cograph with the property that every proper induced subgraph of P4 is a cograph. The main result of the paper identifies non-2-cographs that are minimal in a certain natural sense. In Section 2, we show that 2-cographs can be recursively defined and are closed under taking induced minors. We also generalize cographs to k-cographs for k exceeding two and note that, in that case, the class of k-cographs is not closed under contraction. In addition, we correct a re- sult of Akiyama and Harary [1] that had claimed to characterize when the complement of a 2-connected graph is 2-connected. In Section 3, we show that, in contrast to Theorem 1.1, the class of non- 2-cographs for which every proper induced minor is a 2-cograph is infinite. This prompts us to determine those non-2-cographs whose complements Date: February 28, 2021. 2020 Mathematics Subject Classification. 05C40, 05C83. 1 2 JAMES OXLEY AND JAGDEEP SINGH are also induced-minor-minimal non-2-cographs. The following is the main result of the paper. Its proof occupies most of Section 3. Theorem 1.2. Let G be a graph. Suppose that G is not a 2-cograph but every proper induced minor of each of G and G is a 2-cograph. Then 5 ≤ jV (G)j ≤ 10. The unique 5-vertex graph satisfying the hypotheses of the last theorem is C5, the 5-vertex cycle. In the appendix, we list all of the other graphs that satisfy these hypotheses. 2. Preliminaries Let G be a graph. A vertex u of G is a neighbour of a vertex v of G if uv is an edge of G. The neighbourhood NG(v) of v in G is the set of all neighbours of v in G.A t-cut of G is set Xt of vertices of G such that jXtj = t and G − Xt is disconnected. A graph that has no t-cut for all t less than k is k-connected. Viewing G as a subgraph of Kn where n = jV (G)j, we colour the edges of G green while assigning the colour red to the non-edges of G. In this paper, we use the terms green graph and red graph for G and its complementary graph G, respectively. An edge of G is called a green edge while a red edge refers to an edge of G. The green degree of a vertex v of G is the number of green neighbours of v while the red degree of v is its number of red neighbours. Similarly, a green t-cut refers to a t-cut of the graph G while a red t-cut is a t-cut of the graph G. For graphs G1 and G2 having disjoint vertex sets, the 0-sum G1 ⊕ G2 of G1 and G2 is their disjoint union. A 1-sum G1 ⊕1 G2 of G1 and G2 is obtained by identifying a vertex of G1 with a vertex of G2. We omit the straightforward proofs of the next three results. Lemma 2.1. Let G be a 2-cograph. (i) Every induced subgraph of G is a 2-cograph. (ii) G is a 2-cograph. Lemma 2.2. Let G1 and G2 be 2-cographs and let G be a 0-sum or 1-sum of G1 and G2. Then G is a 2-cograph. Lemma 2.3. Let G be a graph and let uv be an edge e of G. Then the complement of G=e is the graph obtained by adding a vertex w with neigh- bourhood NG(u) \ NG(v) to the graph G − fu; vg. A consequence of the fact that we consider only simple graphs here is that, when we write G=e for an edge e of a graph G, we mean the graph that we get from the multigraph obtained by contracting the edge e and then deleting all but one edge from each class of parallel edges. Lemma 2.4. Let G be a 2-cograph and e be an edge of G. Then G=e is a 2-cograph. 2-COGRAPHS 3 Proof. Assume to the contrary that G=e is not a 2-cograph. Then there is an induced subgraph H of G=e such that both H and H are 2-connected. Let e = uv and let w denote the vertex in G=e obtained by identifying u and v. We may assume that w is a vertex of H, otherwise H is an induced subgraph of G, a contradiction. We assert that the subgraph H0 of G induced on the vertex set (V (H) [ fu; vg) − w is 2-connected as is its complement H0. Since H is 2-connected, H0 is 2-connected unless it has e as a cut-edge. In the exceptional case, we may assume that H0 − u =∼ H. Thus G has an induced subgraph so that both it and its complement are 2-connected, a contradiction. We deduce that H0 is 2-connected. Observe that the neighbours of w in H are the common neighbours of u and v in H0. Thus the degrees of u and v in H0 each equal at least the degree of w in H. Moreover, H0 − u has a subgraph isomorphic to H and is therefore 2-connected. Since u has degree at least two in H0, it follows that H0 is 2-connected, a contradiction. Note that Lemmas 2.1 and 2.4 imply that 2-cographs are closed under tak- ing induced minors. Cographs are also called complement-reducible graphs due to the following recursive-generation result [3]. The operation of taking the complement of a graph is called complementation. Lemma 2.5. A graph G is a cograph if and only if it can be generated from K1 using complementation and 0-sum. Next, we show that if we include the operation of 1-sum in the above operations, we get the class of 2-cographs. Lemma 2.6. A graph G is a 2-cograph if and only if it can be generated from K1 using complementation, 0-sum and 1-sum. Proof. Let G be a 2-cograph. If jGj = 2, the result holds. We proceed via induction on the number of vertices of G and assume that the result holds for all 2-cographs of order less than jGj. Since G is a 2-cograph, either G or G is not 2-connected. Without loss of generality, we may assume that G is not 2-connected and can therefore be written as a 0-sum or 1-sum of two induced subgraphs G1 and G2 of G. By Lemma 2.1, G1 and G2 are 2-cographs and the result follows by induction. Conversely, let G be a graph that can be generated from K1 using com- plementation, 0-sum and 1-sum. Since K1 is a 2-cograph, the result follows by Lemmas 2.1 and 2.2. The following recursive-generation result for cographs is due to Royle [10]. It uses the concept of join of two disjoint graphs G and H, which is the graph G 5 H obtained by joining every vertex of G to every vertex of H. Lemma 2.7. Let C be the class of graphs defined as follows: (i) K1 is in C; (ii) if G and H are in C, then so is their 0-sum; and 4 JAMES OXLEY AND JAGDEEP SINGH (iii) if G and H are in C, then so is their join. Then C is the class of cographs. We generalise the join operation to 1-join G 51 H of graphs G and H obtained by joining every vertex of V (G) − V (H) to every vertex of V (H) − V (G) where jV (G) \ V (H)j is allowed to be non-zero but is at most one. Note that G 51 H is the graph G ⊕ H if G and H are disjoint. If G and H have a common vertex v, then G 51 H = G ⊕1 H where the 1-sum of G and H is obtained by identifying the common vertex v.
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