On the structure of dominating graphs
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1 On the structure of dominating graphs Saeid Alikhani a, Davood Fatehi a Sandi Klavžar b,c,d December 23, 2015 a Department of Mathematics, Yazd University, , Yazd, Iran alikhani@yazd.ac.ir, davidfatehi@yahoo.com b Faculty of Mathematics and Physics, University of Ljubljana, Slovenia sandi.klavzar@fmf.uni-lj.si c Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia d Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia Abstract The k-dominating graph D k (G) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. In this paper, after presenting several basic properties of k-dominating graphs, it is proved that if G is a graph with no isolates, of order n 2, and with G = D k (G), then k = 2 and G = K 1,n 1 for some n 4. The problem of which graphs are dominating graphs is also considered. In particular it is shown that paths and cycles of order at least 5 are not dominating graph. Some results on the order of dominating graphs are also obtained. Keywords: domination; dominating graph; paths and cycles; domination polynomial AMS Subj. Class.: 05C69, 05C62 1 Introduction In this paper, we continue the study of k-dominating graphs, a concept recently introduced by Haas and Seyffarth [11] as follows. If G is a graph, then the k-dominating graph D k (G) of G is the graph whose vertices correspond to the dominating sets of G that have cardinality at most k, two vertices of D k (G) being adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. We point out that a similar concept is the one of γ-graphs in which only minimum dominating sets are considered as vertices of the derived graph [10]. Corresponding author 1
2 The introduction of the k-dominating graphs was motivated by similar studied on graph colorings, by a reconfiguration problem (which asks whether a feasible solution to a problem can be transformed into another by some allowable set of moves, while maintaining feasibility at all steps), and in general to further understand the relationships between dominating sets of a graph. For instance, given dominating sets S and T, is there a sequence of dominating sets S = S 1, S 2,..., S k = T such that each S i+1 is obtained from S i by deleting or adding a single vertex? A natural first problem about k-dominating graphs is to determine conditions that ensure that D k (G) is connected. This problem was the central theme of the seminal paper [11]. It is interesting to observe that the connectedness of D k (G) does not guarantee the connectedness of D k+1 (G). For instance, D k (K 1,n 1 ) (n 4) is connected for any 1 k n 2, but D n 1 (K 1,n 1 ) is not connected. For the latter fact note that Γ(K 1,n 1 ) = n 1 and that in general D Γ(G) (G) is not connected. (Here Γ(G) is the upper domination number of G, that is, the maximum cardinality of a minimal dominating set of G.) On the other hand, Haas and Seyffarth proved that if G has at least two disjoint edges and k min{n 1, Γ(G) + γ(g)}, then D k (G) is connected. Moreover, if G is bipartite or chordal, then D Γ(G)+1 (G) is always connected. We proceed as follows. In the next section, we introduce additional concepts needed, recall some basic properties of k-dominating graphs, and add additional results to this list. In Section 3, we attack the question from [11] where it was observed that D 2 (K 1,n ) = K 1,n and asked whether there are other graphs G for which D k (G) = G holds. We prove that if G is of order n 2 and with δ 1, and if G = D k (G), where γ(g) k n, then actually G = K 1,n 1 holds for some n 4. We also show that paths and cycles of order at least 5 are not k-dominating graphs. In the final section we present some results on the order of k-dominating graphs. Along the way several problems for further study are also listed. 2 Preliminaries We use the notation [n] = {1,..., n}. As usual, δ(g) and (G) denote the minimum and the maximum degree of a graph G, respectively. The order of a graph G = (V, E) will be denoted G, that is, G = V. The disjoint union of graphs G and H will be denoted with G H. The join G + H of graphs G and H is obtained from the disjoint union of G and H by connecting any vertex of G with any vertex of H. If G = (V, E) is a graph, then S V is a dominating set of G if every vertex in V S is adjacent to at least one vertex in S. The domination number γ(g) of G is the minimum cardinality of a dominating set in G. A dominating set of the minimum cardinality is called a γ-set. A vertex of G of degree G 1 is called a dominating vertex of G. For additional concepts from the domination theory see [12]. We say that a graph is a dominating graph if it isomorphic to D k (G) for some graph G and some positive integer k. For example, C 6 is a dominating graph because D 2 (K 3 ) = C 6. In the next result we collect several basic properties about dominating graphs. 2
3 Proposition 2.1 If G is a graph, then the following hold. (i) If γ(g) k G, then D k (G) is bipartite. (ii) D G (G) is odd and D G 1 (G) is even. (iii) If m is odd, 0 < m < 2n, then there exists a graph X of order n such that D n (X) = m. (iv) If G is connected, then (D G (G)) = G. Proof. (i) Note that D n (K n ) is isomorphic to the graph obtained from the n-cube Q n by deleting one of its vertices. Since D k (G) is a subgraph of D k (K n ) and the latter graph is a subgraph of the bipartite graph D n (K n ), it follows that D k (G) is bipartite. (ii) That the order of D G (G) is odd follows immediately from a result of Brouwer, Csorba, and Schrijver [7, Theorem 1.1] asserting that the number of dominating sets of a finite graph is odd. As the only dominating set of order n of G is its vertex set, D G 1 (G) is then of even order. (iii) This assertion follows from [7, Proposition 1.2] which asserts that if m is odd, where 0 < m < 2n, then there exists a graph of order n that contains precisely m dominating subsets (see also [1]). (iv) As D G (G) is a subgraph of Q G we infer that (D G (G)) G. On the other hand, since G is connected, any ( G 1)-subset of vertices is a dominating set and adjacent to the whole vertex set in D G (G). So V (G) is of degree G in D G (G). D G (G) is not regular unless G is an edge-less graph in which case D G (G) = K 1. Note also that from Proposition 2.1(i) and (ii) it follows that D G (G) is not hamiltonian. On the other hand, the question which k-dominating graphs D k (G) with k < G are hamiltonian remains as open problem. 3 Realizability of graphs as dominating graphs Haas and Seyffarth [11] observed that D 2 (K 1,n ) = K 1,n and posed the question whether there are other graphs G for which D k (G) = G. In the next result we prove that the answer is negative as soon as G has no isolated vertices. Theorem 3.1 Let G be a graph of order n 2 and with δ 1. If G = D k (G), where γ(g) k n, then k = 2 and G = K 1,n 1 for some n 4. Proof. Assume first that k = γ(g). Then V (D k (G)) consists of γ-sets of G and hence D k (G) is an edge-less graph. If G = D k (G), this is only possible when G = K 1. As we have assumed that G is of order at least 2, we may suppose in the rest of the proof that k γ(g) + 1. Let V (G) = {v 1,..., v n } and set γ = γ(g). Let X be a γ-set of G, where we may without loss of generality assume that X = {v 1,..., v γ }. Assume that G = D k (G) and recall that k γ + 1. Then X i = X {v i }, γ + 1 i n, are dominating sets 3
4 of G and hence vertices of D k (G). As they are all of cardinality γ + 1, the vertices X, X γ+1,..., X n induce a K 1,n γ in D k (G) (hence G also contains an induced K 1,n γ ). Let Y = {Y 1,..., Y γ 1 } be the remaining vertices of D k (G). Observe that the vertex Y i, i [γ 1], is not adjacent to X, for otherwise {X} {Y j : j i} would be a dominating set of D k (G), but then (since G = D k (G)) we would have a dominating set of G smaller than γ. Moreover, a vertex X i, γ + 1 i n, can be adjacent to at most one vertex from Y. Indeed, suppose that, without loss of generality, X γ+1 is adjacent to Y 1 and Y 2. Then {X, X γ+1, Y 3,..., Y γ 1 } is a dominating set D k (G) yielding the same contradiction as above. By the theorem s assumption, G has no isolated vertices. As each Y i has a neighbor in X = {X γ+1,..., X n } and no X j is adjacent to two vertices from Y, we find out that there exists a matching from Y to X. Let {X i1,..., X iγ 1} be the endpoints of the matching edges which lie in X. Then X, X i1,..., X iγ 1 is a dominating set of G of cardinality γ and we have the following two γ-sets of D k (G): X, Y 1,..., Y γ 1 and X, X i1,..., X iγ 1. Suppose that γ 2. Adding to any of the above two γ-sets an additional vertex, we get a dominating set of cardinality γ + 1. Since γ 2, in this way we can construct 2(n γ) 1 different dominating sets of G of this cardinality. Consequently, D k (G) contains at least 2 + 2(n γ) 1 vertices. Since for any graph without isolated vertices γ n/2 holds, it follows that D k (G) contains at least n + 1 vertices, a contradiction. The only case left to consider is γ = 1. Assume without loss of generality that v 1 is a dominating vertex. Suppose that G contains another dominating vertex, say v 2, that is, deg(v 1 ) = deg(v 2 ) = n 1. Then {v 1 }, {v 2 }, {v 1, v 2 }, {v 1, v i } (i 3), and {v 2, v i } (i 3), are dominating sets of G, hence D 2 (G) 2n 1 > n. Therefore v 1 is the unique dominating vertex of G. Now, since D 2 (G) is of order n, its dominating sets of order at most 2 are {v 1 } and {v 1, v i } (i 2). But then D 2 (G) = K 1,n 1 where n 4. Let G be an arbitrary graph with γ(g) 3 and consider the join G + K 1. Clearly, γ(g + K 1 ) = 1. Moreover, if D is a dominating set of G + K 1 and D = 2, then (since γ(g) 3) we must have x D. It follows that D 2 (G + K 1 ) = K 1, G. This example shows, as a counterpart to Theorem 3.2, that stars K 1,n can be represented as a dominating graph in many different ways. Another problem from [11] is which graphs are dominating graphs. In the next result we give a rather restrictive condition on the order of connected graphs whose dominating graphs are regular. To state the result, a short preparation is needed. For any r 1 let c r be a given, fixed constant such that γ(g) c r G holds for any connected graph G with δ(g) = r. As already observed by Ore [14], if δ(g) 1 then γ(g) n/2. Hence for any r we can select c r such that 0 < c r 1/2 holds. 4
5 Theorem 3.2 If G is a connected graph such that D k (G) is connected and r-regular (k, r 1), then r G. 1 c δ(g) Proof. Suppose that D k (G) = H, where H is an r-regular graph and k is a positive integer. Let X be a γ-set of G. Then for any vertex y / X, the set X y = X {y} is a dominating set of order γ(g) + 1. Hence X y is a neighbor of X in D k (G). Because there are G γ(g) such vertices y, we infer that deg Dk (G)(X) = G γ(g). As D k (G) = H and H is r-regular it follows that r = G γ(g). Since γ(g) c δ(g) G we have G r = γ(g) c δ(g) G which immediately yields the claimed inequality. Corollary 3.3 Among the cycles, C 6 and C 8 are the only dominating graphs of connected graphs. Proof. As already remarked, for any r we can select c r such that c r 1/2 holds. Now, if G is connected and D k (G) = C m for a graph G and m 3, Theorem 3.2 implies that G 4. By inspection we find out that among such connected graphs the only favourable cases are D 2 (K 3 ) = C 6 and D 3 (P 4 ) = C 8, see Fig. 1 for the latter case Figure 1: P 4 and D 3 (P 4 ) In order to further apply Theorem 3.2, it is useful to know the best known constants c r. The constant c 2 = 3/5 was independently obtained in [5, 13] (actually, there are seven small graphs: C 4, and six graphs on seven vertices, for which the 3/5 bound does not hold); the result c 3 = 3/8 is due to Reed [15]; c 4 = 4/11 is from [16]. For k 5 the best known constants c k were recently developed in [9]. To obtain these constants a modification of a method from [8] was applied, which was in turn developed for the investigation of the domination game [6]. A result parallel to Corollary 3.3 for paths reads as follows. Proposition 3.4 Among the paths, P 1 and P 3 are the only dominating graphs of connected graphs. Proof. By inspection on connected graphs of order at most 4 the only dominating graphs that are paths are P 1 = D1 (K 1 ) and P 3 = D2 (P 2 ). Suppose now that D k (G) = P m holds for some connected graph G with G > 4 and for some k and m. Let X be a γ-set of G. Then either deg Dk (G)(X) = 1 or 5
6 deg Dk (G)(X) = 2. Since clearly k > γ(g), it follows that either G γ(g) = 1 or G γ(g) = 2. But this is not possible since G > 4. In Corollary 3.3 and in Proposition 3.4 we have considered the dominating graphs that are derived from connected graphs. The following examples indicate that it would be interesting to extend the investigation to disconnected graphs: D 3 (K 2 K 2 ) = C 8, D 3 (K 3 K 2 ) = C 6 C 6, and D 3 (K 2 K 1 ) = D 4 (K 2 K 1 K 1 ) = P 3. Similarly, in Theorem 3.2 we have assumed that the graph G considered has no isolated vertices, hence an extension to graphs that contain isolates could also be interesting. Paths and cycles of order greater than 4 are thus not dominating graphs. On the other hand, a characterization of dominating graphs is widely open. 4 On the order of dominating graphs The domination polynomial D(G, x) of G is defined as D(G, x) = i 0 d(g, i)x i, where d(g, i) is the number of dominating sets of G of cardinality i. This graph polynomial was introduced in the paper [3] that appeared in 2014 but numerous other papers on the polynomial appeared earlier. For some very recent developments on the polynomial see [4]. From our perspective, a key information encoded into the domination polynomial is that D G (G) = D(G, 1). For instance, using a result from [2] asserting that D(C 1, x) = x, D(C 2, x) = x 2 + 2x, D(C 3, x) = x 3 + 3x 2 + 3x, and D(C n, x) = x (D(C n 1, x) + D(C n 2, x) + D(C n 3, x)) for n 4, we get the following result. Proposition 4.1 D 1 (C 1 ) = 1, D 2 (C 2 ) = 3, D 3 (C 3 ) = 5, and D n (C n ) = D n 1 (C n 1 ) + D n 2 (C n 2 ) + D n 3 (C n 3 ), n 4. We conclude the paper by determining the order of the dominating graph of the join and the corona of two graphs in terms of the invariants of their factors. The join has already been defined, while the corona G H of graphs G and H is the graph obtained from the disjoint union of G and G copies of H by joining the ith vertex of G (1 i G ) to every vertex in the i-th copy of H. Proposition 4.2 If G and H are graphs, then (i) D G+H (G + H) = (2 G 1)(2 H 1) + D G (G) + D H (H), (ii) D G H (G H) = (2 H + D H (H) ) G. 6
7 Proof. (i) Note first that any set D V (G + H) such that D V (G) and D V (H) is a dominating set of G + H. This gives (2 G 1)(2 H 1) dominating sets of G + H. Assume now that D is a dominating set of G + H with D V (G) =. Then D V (H) must be a dominating set of H, whence there are D H (H) such dominating sets of G + H. Analogously, if D V (H) = we get D G (G) dominating sets of G + H. (ii) Let D be a dominating set of G H and assume that a vertex x V (G) does not belong to D. If H x is the copy of H corresponding to x, then D V (H x ) is a dominating set of D x. Therefore, if a vertex x V (G) is not in D, it is dominated by a vertex from H x. It follows that D is a dominating set of G H if and only if D is a dominating set of the graph (G H) E(G). The latter graph is isomorphic to the disjoint union of G copies of the graph K 1 + H. By (i), D H +1 (K 1 + H) = 2 H + D H (H) and consequently D G H (G H) = (2 H + D H (H) ) G. References [1] S. Alikhani, The domination polynomial of a graph at 1, Graphs Combin. 29 (2013) [2] S. Alikhani, Y.H. Peng, Dominating sets and domination polynomials of certain graphs, II, Opuscula Math. 30 (2010) [3] S. Alikhani, Y.H. Peng, Introduction to domination polynomial of a graph, Ars Combin. 114 (2014) [4] B.M. Anthony, M.E. Picollelli, Complete r-partite graphs determined by their domination polynomial, Graphs Combin. 31 (2015) [5] M.M. Blank, An estimate of the external stability number of a graph without suspended vertices (in Russian), Prikl. Mat. i Programmirovanie 10 (1973) [6] B. Brešar, S. Klavžar, D. F. Rall, Domination game and an imagination strategy, SIAM J. Discrete Math. 24 (2010) [7] A.E. Brouwer, P. Csorba, A. Schrijver, The number of dominating sets of a finite graph is odd, preprint (June 2, 2009), domin4a.pdf. [8] Cs. Bujtás, Domination game on forests, Discrete Math. 338 (2015) [9] Cs. Bujtás, S. Klavžar, Improved upper bounds on the domination number of graphs with minimum degree at least five, Graphs Combin., to appear, DOI: /s [10] G. Fricke, S.M. Hedetniemi, S.T. Hedetniemi, K.R. Hutson, γ-graphs of graphs, Discuss. Math. Graph Theory 31 (2011)
8 [11] R. Haas, K. Seyffarth, The k-dominating graph, Graphs Combin. 30 (2014) [12] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, [13] W. McCuaig, B. Shepherd, Domination in graphs with minimum degree two, J. Graph Theory 13 (1989) [14] O. Ore, Theory of Graphs, American Mathematical Society, Providence, R.I., [15] B. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) [16] M.Y. Sohn, Y. Xudong, Domination in graphs of minimum degree four, J. Korean Math. Soc. 46 (2009)
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