A characterization of diameter-2-critical graphs with no antihole of length four

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1 Cent. Eur. J. Math. 10(3) DOI: /s x Central European Journal of Mathematics A characterization of diameter-2-critical graphs with no antihole of length four Research Article Teresa W. Haynes 1, Michael A. Henning 2 1 Department of Mathematics, East Tennessee State University, Johnson City, TN , USA 2 Department of Mathematics, University of Johannesburg, Auckland Park, South Africa Received 1 December 2011; accepted 19 January 2012 Abstract: A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n 2 /4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty Simon Conjecture for graphs with no antihole of length four. MSC: 05C12, 05C35, 05C69 Keywords: Diameter critical Diameter-2-critical Antihole Total domination critical Versita Sp. z o.o. 1. Introduction Distance and diameter are fundamental concepts in graph theory. A graph G is called diameter-2-critical if its diameter is two, and the deletion of any edge increases the diameter. Diameter-2-critical graphs are extensively studied in the literature. See, for example, [1 5, 8, 9, 13, 14] and elsewhere. Plesník [14] observed that all known diameter-2-critical graphs on n vertices have no more than n 2 /4 edges and that the extremal graphs appear to be the balanced complete bipartite graphs (complete bipartite graphs with equal size partite sets). Murty and Simon [2] independently made the following conjecture: haynes@etsu.edu mahenning@uj.ac.za 1125

2 A characterization of diameter-2-critical graphs with no antihole of length four Conjecture 1.1. If G is a diameter-2-critical graph with order n and size m, then m n 2 /4, with equality if and only if n is even and G is the complete bipartite graph K n/2,n/2. According to Füredi [5], Erdős said that this conjecture goes back to the work of Ore in the 1960s. Plesník [14] proved that m < 3n(n 1)/8. Caccetta and Häggkvist [2] showed m <.27n 2. Fan [4] proved the first part of the conjecture for n 24 and for n = 26. For n 25, he obtained m < n 2 /4 + (n n + 56)/320 <.2532n 2. Then Xu [15] gave an incorrect proof of the conjecture in In 1992 Füredi [5] gave an asymptotic result proving the conjecture is true for large n, that is, for n > n 0 where n 0 is a tower of 2 s of height about Recently in [8] the conjecture was proved true for graphs whose complements have diameter 3, while in [9] it was proved true for graphs whose complements have no induced claw. Although considerable work has been done in an attempt to completely resolve the conjecture and several impressive partial results have been obtained, the conjecture remains open for general n. We say that a graph is F-free if it does not contain F as an induced subgraph. We denote a path on n vertices by P n and a cycle on n vertices by C n. If G is C k -free for some k 4, then we say that G has no hole of length k and equivalently its complement G has no antihole of length k. 2. Main results Our aim in this paper is to characterize the diameter-2-critical graphs with no antihole of length 4. For this purpose, we let F denote the family of graphs that can be obtained from a 5-cycle v 1 v 2 v 3 v 4 v 5 v 1 by replacing each vertex v i, 1 i 5, with a nonempty independent set A i and adding all edges between A i and A i+1, where addition is taken modulo 5. A graph in the family F is illustrated in Figure 1. Figure 1. A graph in the family F. We shall prove Theorem 2.1. Let G be a graph with no antihole of length 4. Then, G is a diameter-2-critical graph if and only if G is a complete bipartite graph with minimum degree at least 2 or G F. A proof of Theorem 2.1 is given in Section 3. As an application of our characterization, the Murty Simon Conjecture holds for graphs with no antihole of length 4. A proof of Theorem 2.2 is given in Section 4. Theorem 2.2. Conjecture 1.1 is true for graphs with no antihole of length

3 T.W. Haynes, M.A. Henning 2.1. Graph theory terminology For notation and graph theory terminology, we in general follow [7]. Specifically, let G be a graph with vertex set V (G) = V of order V = n and size E(G) = m, and let v be a vertex in V. The open neighborhood of v is N(v) = {u V : uv E(G)} and the closed neighborhood of v is N[v] = {v} N(v). For a set S V, its open neighborhood is the set N(S) = v S N(v), and its closed neighborhood is the set N[S] = N(S) S. For notational convenience, if u and v are adjacent vertices in G, we write u v, while if u and v are not adjacent, we write u v. For sets S, X V, if X N[S] (X N(S), respectively), we say that S dominates X, written S X (S totally dominates X, respectively, written S t X). If S = {s} or X = {x}, we also write s X, S t x, etc. If S V (S t V, respectively), we say that S is a dominating set (total dominating set) of G, and we also write S G (S t G, respectively). An S-external private neighbor of a vertex v S is a vertex u V \ S which is adjacent to v but to no other vertex of S. The set of all S-external private neighbors of v S is called the S-external private neighbor set of v and is denoted epn(v, S). For a set S V, the subgraph induced by S is denoted by G[S]. If H is a subgraph of G, then we say that H is a dominating H in the graph G if V (H) is a dominating set in G. We remark that if H is a dominating H in G, then the subgraph H of G is not necessarily an induced subgraph of G. If X and Y are two subsets of V, then we denote the set of all edges of G that join a vertex of X and a vertex of Y by [X, Y ]. Further, if all edges are present between the vertices in X and the vertices in Y, we say that [X, Y ] is full, while if there are no edges between the vertices in X and the vertices in Y, we say that [X, Y ] is empty. 3. Proof of Theorem 2.1 In order to prove our main result, we use an important association with total domination in graphs. Conventionally the diameter of a disconnected graph is considered to be either undefined or defined as infinity. For the purposes of this paper, we use the former and hence require that a diameter-2-critical graph have minimum degree at least two (since removing an edge incident to a vertex of degree one results in a disconnected graph). We note however that if we relaxed the condition and defined the diameter of disconnected graphs to be infinity, the only additional diameter-2- critical graphs are stars with order at least three. Under this definition, we can drop the minimum degree requirement on the complete bipartite graphs of Theorem 2.1, and both it and the Murty Simon Conjecture hold The association with total domination A total dominating set, denoted TDS, of G is a set S of vertices of G such that every vertex is adjacent to a vertex in S. Every graph without isolated vertices has a TDS, since S = V is such a set. The total domination number γ t (G) is the minimum cardinality of a TDS. A TDS of G of cardinality γ t (G) is called a γ t (G)-set. Total domination is now well studied in graph theory. For more details, the reader is referred to the domination book [7] and a recent survey on total domination [10]. As introduced in [12], a graph G is total domination edge critical if γ t (G + e) < γ t (G) for every edge e E(G). Further if γ t (G) = k, then we say that G is a k t -critical graph. Thus if G is k t -critical, then its total domination number is k and the addition of any edge decreases the total domination number. It is shown in [12] that the addition of an edge to a graph can change the total domination number by at most two. Total domination edge critical graphs G with the property that γ t (G) = k and γ t (G + e) = k 2 for every edge e E(G) are called k t -supercritical graphs. Hanson and Wang [6] were the first to observe the following key relationship between diameter-2-critical graphs and total domination edge critical graphs. Note that this relationship is contingent on total domination being defined in the complement G of the diameter-2-critical graph G, that is, G has no isolated vertices. Hence our elimination of stars as diameter-2-graphs is necessary for this association. Theorem 3.1 ([6]). A graph is diameter-2-critical if and only if its complement is 3 t -critical or 4 t -supercritical. 1127

4 A characterization of diameter-2-critical graphs with no antihole of length four The 4 t -supercritical graphs are characterized in [11]. Theorem 3.2 ([11]). A graph G is 4 t -supercritical if and only if G is the disjoint union of two complete graphs each of order at least 2. Let G denote the family of graphs that can be obtained from a 5-cycle v 1 v 2 v 3 v 4 v 5 v 1 by replacing each vertex v i, 1 i 5, with a clique A i and adding all edges between A i and A i+1, where addition is taken modulo 5. Thus, G G if and only if G F. By Theorem 3.2, the complement of a 4 t -supercritical graph is a complete bipartite graph with minimum degree at least 2. Therefore by Theorems 3.1 and 3.2, in order to prove our main result, namely Theorem 2.1, it suffices for us to prove the following result, a proof of which is given in subsection 3.3. Theorem 3.3. Let G be a C 4 -free graph. Then, G is a 3 t -critical graph if and only if G G. A chordal graph is a graph with no induced cycles of length more than three. As an immediate consequence of Theorem 3.3, we have the following result. Corollary 3.4. No chordal graph is 3 t -critical. The next corollary follows from Theorems 3.1, 3.2 and 3.3. Corollary 3.5. Complete bipartite graphs with minimum degree at least two are the only diameter-2-critical graphs whose complements are chordal Preliminary observations and results Since γ t (G) 2 for any graph G, the addition of an edge to a 3 t -critical graph reduces the total domination number by exactly one. Hence if G is a 3 t -critical graph, then γ t (G) = 3 and γ t (G + e) = 2 for every edge e E(G). We will frequently use the following observation and notation. Observation 3.1. For every 3 t -critical graph G and nonadjacent vertices u and v in G, either {u, v} dominates G or, without loss of generality, {u, w} dominates G v, but not v, for some w N(u). In this case, we write uw v. If u and v are two nonadjacent vertices in G that do not dominate V, then by Observation 3.1, uw v or vw u for some vertex w V \ {u, v}. Let S be a γ t (G)-set of a 3 t -critical graph G. Then, S = 3 and G[S] is either a P 3 or a K 3. Thus the graph G[S] contains a path P 3 (not necessary induced) that is a dominating P 3 in G. We shall need the following property of a dominating P 3 in a 3 t -critical graph. Lemma 3.6. Let G be a C 4 -free, 3 t -critical graph, and let S = {a, b, c} be a γ t (G)-set. If abc is a dominating P 3 in G, then both epn(a, S) and epn(c, S) induce a clique. 1128

5 T.W. Haynes, M.A. Henning Proof. Let S = {a, b, c} be a γ t (G)-set where abc is a dominating P 3, and let A = epn(a, S) and C = epn(c, S). The minimality of S implies that A and C. Assume, for the sake of contradiction, that {a 1, a 2 } A but a 1 a 2. Neither b nor c is dominated by {a 1, a 2 }, and so {a 1, a 2 } does not dominate G. By Observation 3.1, we may assume, renaming a 1 and a 2 if necessary, that a 1 x a 2. Since a 1 is adjacent to neither b nor c, we have that x {b, c}. Let c C. If x c, then a 1 c. But then xcc a 1 x is an induced C 4 in G, a contradiction. Hence, x c. Since c is an arbitrary vertex in C, we have that x C. If x a, then abxa 1 a is an induced C 4 in G, a contradiction. Hence, x a. Since {a, x} t G, there is a vertex y not dominated by {a, x}. Since a A and x C {b, c}, we note that y / A C {b, c}. However S is a TDS in G, implying that y b or y c. If y b, then y C, a contradiction. Hence, y b (possibly, y c). Since a 1 x a 2, we have y a 1. But then ybxa 1 y is an induced C 4 in G, a contradiction. Therefore, A induces a clique. Analogously, C induces a clique Proof of Theorem 3.3 For a subset D of vertices in G and for {x, y} D, we define V xy = { v V \ D : N(v) D = {x, y} }. Recall the statement of Theorem 3.3. Theorem 3.3. Let G be a C 4 -free graph. Then, G is a 3 t -critical graph if and only if G G. Proof. Let G be a C 4 -free graph. If G G, then it is a simple exercise to check that G is a 3 t -critical graph. This establishes the sufficiency. To prove the necessity, suppose that G is a 3 t -critical graph. Among all dominating P 3 s in G, let abc be chosen so that epn(a, S) + epn(c, S) is minimized, where S = {a, b, c}. Let D = epn(c, S), E = epn(a, S) and F = epn(b, S). By the minimality of the set S, we note that D 1 and E 1. By Lemma 3.6, the set D induces a clique and the set E induces a clique. Since G is C 4 -free, we have that both [E, F] and [D, F] are empty. We proceed further with a series of claims. Claim A. The following hold in the graph G. (a) [D, E] is full. (b) a c. (c) V ac =. (d) F =. Proof of Claim A. (a) Assume, for the sake of contradiction, that there exist vertices a E and c D such that a c. The vertex b is adjacent to neither a nor c, and so {a, c } does not dominate V. By Observation 3.1, we may assume renaming vertices if necessary, that a x c. Since a is adjacent to neither b nor c, we have that x {b, c}. Further since [E, F] is empty, we have that x F. If x = a, then G[S] is a triangle and the minimality of S implies that F. But a F, contradicting that F = epn(b, S). Hence we may assume that x a. If x a, then abxa a is an induced C 4 in G, a contradiction. Hence, x a. Since a x c and c c, we note that a xc is a dominating P 3. Let S = {a, x, c}. Then, epn(c, S ) = {c } D. Let a epn(a, S ). Since x F and c D, we note that a / D F. Further since x {a, b}, we note that a / {a, b}. If a b, then a bxa a is an induced C 4 in G, a contradiction. Hence, a b, implying that a E \ {a }. Since a is an arbitrary vertex in epn(a, S ), we have that epn(a, S ) E \ {a }. But then a xc is a dominating P 3 such that epn(a, S ) + epn(c, S ) D + E 1, contradicting our choice of the path abc. Therefore, [D, E] is full. (b) Let a E and c D. By (a), a c. If a c, then acc a a is an induced C 4 in G, a contradiction. Hence, a c. 1129

6 A characterization of diameter-2-critical graphs with no antihole of length four (c) If x V ac, then abcxa is an induced C 4 in G, a contradiction. Hence, V ac =. (d) Assume that F. As observed earlier, both [E, F] and [D, F] are empty. Let b F and consider G + ab. Since no vertex in D {c} is dominated by {a, b }, by Observation 3.1, ax b or b x a. Suppose that b x a. Then, x D E {c}. If x b, then xb bcx is an induced C 4 in G, a contradiction. Hence, x b. But then xbaa x, where a E, is an induced C 4 in G, a contradiction. Therefore, ax b, implying that x D {c}. By (c), V ac =, and so x b. If x y for some vertex y E, then axc ya is an induced C 4 in G, where c D, a contradiction. Hence, x E. Since x D E and since V ac =, we have that {x, b} t V, a contradiction. Therefore, F =. We now return to the proof of Theorem 3.3. Among all the vertices of E, let a be one with maximum degree, and similarly, select c from D to have maximum degree among the vertices of D. Let A = N(a ) \ (D E), C = N(c ) \ (D E), B = V \ (A C D E). We note that a A, b B and c C. Also it follows from Claim A (c) that b B. Claim B. The following hold in the graph G. (a) b A C. (b) A induces a clique and C induces a clique. (c) A C =. (d) [A, C] is empty. (e) [A, E] is full and [C, D] is full. (f) Each of the sets [A, D], [B, D], [B, E] and [C, E] is empty. (g) [A, B] is full and [B, C] is full. (h) B induces a clique. Proof of Claim B. (a) Suppose that x A but x b. Then, x a. Since x / D E we have that x V ac, contradicting Claim A (c). Hence, b A. Analogously, b C. (b) Suppose that {a 1, a 2 } A but a 1 a 2. Then, a a 1 ba 2 a is an induced C 4 in G, a contradiction. Hence, A induces a clique. Analogously, C induces a clique. (c) Suppose that x A C. Then, x a and x c. In particular, x / {a, c}. By (a), we have that x b, and by (b), x a and x c. Since {x, b} t V, there exists a vertex y not dominated by {x, b}. By Claim A (c), V ac =, and so y D E. By symmetry, we may assume that y E. But then yaxc y is an induced C 4 in G, a contradiction. Therefore, A C is empty. (d) Suppose that x y, where x A and y C. Then, a xyc a is an induced C 4 in G, a contradiction. Therefore, [A, C] is empty. (e) Suppose that x y, where x E and y A. Since x a, we note that y a. Since {x, y} does not dominate the vertex c, by Observation 3.1, xw y or yw x. Suppose that xw y. Now w {b, c}, and so w D E. Further w A since A induces a clique and y A. If w c, then wcc xw is an induced C 4 in G. Hence w c and by the definition of C, w C. Part (d) implies that w a. But now abwxa is an induced C 4, a contradiction. Therefore, yw x. In particular, w x. Since x E, and since E induces a clique and [D, E] is full, the vertex w / D E. By (d), the set [A, C] is empty, and so since w y and y A, we note that w / C. But then neither w nor y dominate the vertex c, a contradiction. Therefore, [A, E] is full. Analogously, [C, D] is full. (f) By Lemma 3.6, each of the sets D and E induces a clique. By (b), C induces a clique, and by (e), [C, D] is full. Hence, N[c ] = C D E N[d] for every vertex d D. Therefore by our choice of the vertex c, we have that N[d] = C D E 1130

7 T.W. Haynes, M.A. Henning for all d D. Thus [A, D] is empty and [B, D] is empty. Analogously, by our choice of a, N[x] = A D E for every vertex x E, implying that [B, E] is empty and [C, E] is empty. (g) Suppose that x y, where x A and y B. Since b A, we note that b y. Since b B, we have that b y. Part (f) implies that c is not dominated by {x, y}. Therefore by Observation 3.1, xw y or yw x. Suppose that xw y. In particular, we note that w x, and so w A B E since [A, C D] is empty. If w A B, then the vertex c is not dominated by {x, w}. If w E, then no vertex in C is dominated by {x, w}. Both cases produce a contradiction. Hence yw x. In particular, we note that w x, and so w A E. By (f), [B, D] is empty, and so since w y, we have w D. But then {w, y} E, a contradiction. Therefore, [A, B] is full. Analogously, [B, C] is full. (h) Suppose that {b 1, b 2 } B but b 1 b 2. By (g), [A, B] is full and [B, C] is full. Hence, ab 1 cb 2 a is an induced C 4 in G, a contradiction. Therefore, B induces a clique. We now return to the proof of Theorem 3.3. By construction, the nonempty sets A, B, C, D and E form a partition of the vertex set V. By Lemma 3.6, the set D induces a clique and the set E induces a clique. By Claim B, each of the sets A, B and C induces a clique. By Claim A (a), the set [D, E] is full. By Claim B, all the sets [A, B], [B, C], [C, D] and [A, E] are full. Again by Claim B, each of the sets [A, C], [A, D], [B, D], [B, E] and [C, E] is empty. Thus, G G. This completes the proof of Theorem Proof of Theorem 2.2 As an application of our main characterization, we prove in this section that Conjecture 1.1 is true for graphs with no antihole of length 4. We shall need the following observation. Observation 4.1. If i + j = n for nonnegative integers i and j, then ij = n2 4 (i j)2 4 Corollary 4.1. Let G be a graph of order n and size m. If n = i + j and m ij, then m n 2 /4 with strict inequality if i j.. If u and v are two nonadjacent vertices in a graph G, then we say that uv is a missing edge in G. Corollary 4.1 implies that Conjecture 1.1 is true for bipartite diameter-2-critical graphs. In particular, the conjecture bound is sharp for complete balanced bipartite graphs with minimum degree at least two, and strict inequality of the bound is reached for bipartite graphs G with minimum degree at least two if the nontrivial partite sets of G have different size, or if the partite sets of G have the same size and G has at least one missing edge between vertices of different partite sets (that is, G is an incomplete balanced bipartite graph). For the purpose of applying Corollary 4.1 to count edges, we say that a set S of k vertices forms a pseudo-independent set if we can uniquely associate ( k 2) missing edges of G with S. As a consequence of Corollary 4.1 we have the following result. Corollary 4.2. Let G be a diameter-2-critical graph. Conjecture 1.1 holds if G is a complete bipartite graph or if V (G) can be partitioned into two pseudo-independent sets of different size or into two pseudo-independent sets of the same size with at least one additional missing edge not associated with either pseudo-independent set. Recall the statement of Theorem

8 A characterization of diameter-2-critical graphs with no antihole of length four Theorem 2.2. Conjecture 1.1 is true for graphs with no antihole of length 4. Proof. Let G be a diameter-2-critical graph with no antihole of length 4. If G is a complete bipartite graph with minimum degree at least two, then Conjecture 1.1 is true. Hence we may assume that G is not a complete bipartite graph with minimum at least degree two. By Theorem 2.1, G F. Hence we can partition V (G) into five nonempty independent sets A, B, C, D, E such that each set [A, B], [B, C], [C, D], [D, E] and [A, E] is full and each set [A, C], [A, D], [B, D], [B, E] and [C, E] is empty. Let A = a, B = b, C = c, D = d and E = e. We say that two sets X and Y in {A, B, C, D, E} are non-consecutive in G if [X, Y ] is empty. Renaming the sets if necessary, we may assume that among all non-consecutive sets in {A, B, C, D, E}, the sets A and C are such that a + c is a maximum. Thus, c d and a e. Further we may assume without loss of generality that c a. Hence, ad + ce ad + ae = a(d + e) e(d + e) = de + e 2 > de, and so the missing ad edges in [A, D] together with the missing ce edges in [C, E] strictly exceeds [D, E]. Therefore we can uniquely associate de missing edges in [A, D] [C, E] with the set [D, E]. Further, ad + ce de 1 missing edges in [A, D] [C, E] are not associated with any missing edges of G in [D, E]. Hence, V can be partitioned into two pseudo-independent sets A C and B D E with at least one additional missing edge not associated with either pseudo-independent set. Therefore by Corollary 4.2, Conjecture 1.1 holds when G has no antihole of length 4. Acknowledgements The second author is supported in part by the University of Johannesburg and the South African National Research Foundation. References [1] Bondy J.A., Murty U.S.R., Extremal graphs of diameter two with prescribed minimum degree, Studia Sci. Math. Hungar., 1972, 7, [2] Caccetta L., Häggkvist R., On diameter critical graphs, Discrete Math., 1979, 28(3), [3] Chen Y.-C., Füredi Z., Minimum vertex-diameter-2-critical graphs, J. Graph Theory, 2005, 50(4), [4] Fan G., On diameter 2-critical graphs, Discrete Math., 1987, 67(3), [5] Füredi Z., The maximum number of edges in a minimal graph of diameter 2, J. Graph Theory, 1992, 16(1), [6] Hanson D., Wang P., A note on extremal total domination edge critical graphs, Util. Math., 2003, 63, [7] Haynes T.W., Hedetniemi S.T., Slater P.J., Fundamentals of Domination in Graphs, Monogr. Textbooks Pure Appl. Math., 208, Marcel Dekker, New York, 1998 [8] Haynes T.W., Henning M.A., van der Merwe L.C., Yeo A., On a conjecture of Murty and Simon on diameter 2-critical graphs, Discrete Math., 2011, 311(17), [9] Haynes T.W., Henning M.A., Yeo A., A proof of a conjecture on diameter 2-critical graphs whose complements are claw-free, Discrete Optim., 2011, 8(3), [10] Henning M.A., A survey of selected recent results on total domination in graphs, Discrete Math., 2009, 309(1), [11] van der Merwe L.C., Total Domination Critical Graphs, PhD thesis, University of South Africa, 1998 [12] van der Merwe L.C., Mynhardt C.M., Haynes T.W., Total domination edge critical graphs, Util. Math., 1998, 54, [13] Murty U.S.R., On critical graphs of diameter 2, Math. Mag., 1968, 41, [14] Plesník J., Critical graphs of given diameter, Acta Fac. Rerum Natur. Univ. Comenian. Math., 1975, 30, (in Slovak) [15] Xu J.M., Proof of a conjecture of Simon and Murty, J. Math. Res. Exposition, 1984, 4, (in Chinese) 1132