On the Super Domination Number of Graphs

On the super domination number of graphs Douglas J. Klein1, Juan A. Rodr´ıguez-Velázquez1,2, Eunjeong Yi1 1Texas A&M University at Galveston Foundational Sciences, Galveston, TX 77553, United States 2Universitat Rovira i Virgili, Departament d’Enginyeria Informàtica i Matemàtiques Av. Pa¨ısos Catalans 26, 43007 Tarragona, Spain Email addresses: [email protected], [email protected],[email protected] April 24, 2018 Abstract The open neighbourhood of a vertex v of a graph G is the set N(v) consisting of all vertices adjacent to v in G. For D ⊆ V (G), we define D = V (G) \ D. A set D ⊆ V (G) is called a super dominating set of G if for every vertex u ∈ D, there exists v ∈ D such that N(v) ∩ D = {u}. The super domination number of G is the minimum cardinality among all super dominating sets in G. In this article, we obtain closed formulas and tight bounds for the super domination number of G in terms of several invariants of G. Furthermore, the particular cases of corona product graphs and Cartesian product graphs are considered. Keywords: Super domination number; Domination number; Cartesian product; Corona product. AMS Subject Classification numbers: 05C69; 05C70 ; 05C76 1 Introduction arXiv:1705.00928v2 [math.CO] 23 Apr 2018 The open neighbourhood of a vertex v of a graph G is the set N(v) consisting of all vertices adjacent to v in G. For D ⊆ V (G), we define D = V (G) \ D. A set D ⊆ V (G) is dominating in G if every vertex in D has at least one neighbour in D, i.e., N(u) ∩ D =6 ∅ for every u ∈ D. The domination number of G, denoted by γ(G), is the minimum cardinality among all dominating sets in G. A dominating set of 1 cardinality γ(G) is called a γ(G)-set. The reader is referred to the books [9, 10] for details on domination in graphs. A set D ⊆ V (G) is called a super dominating set of G if for every vertex u ∈ D, there exists v ∈ D such that N(v) ∩ D = {u}. (1) If u and v satisfy (1), then we say that v is a private neighbour of u with respect to D. The super domination number of G, denoted by γsp(G), is the minimum cardinality among all super dominating sets in G. A super dominating set of cardinality γsp(G) is called a γsp(G)-set. The study of super domination in graphs was introduced in [14]. We recall some results on the extremal values of γsp(G). Theorem 1. [14] Let G be a graph of order n. The following assertions hold. • γsp(G)= n if and only if G is an empty graph. n • γsp(G) ≥ ⌈ 2 ⌉. ∼ ∼ • γsp(G)=1 if and only if G = K1 or G = K2. n It is well known that for any graph G without isolated vertices, 1 ≤ γ(G) ≤ ⌈ 2 ⌉. As noticed in [14], from the theorem above we have that for any graph G without isolated vertices, n 1 ≤ γ(G) ≤ ≤ γ (G) ≤ n − 1. (2) 2 sp l n m Connected graphs with γsp(G) = 2 were characterized in [14], while all graphs with γsp(G)= n − 1 were characterized in [4]. The following examples were previously shown in [14]. (a) For a complete graph Kn with n ≥ 2, γsp(Kn)= n − 1. (b) For a star graph K1,n−1, γsp(K1,n−1)= n − 1. (c) For a complete bipartite graph Kr,t with min{r, t} ≥ 2, γsp(Kr,t)= r + t − 2. The three cases above can be generalized as follows. Let Ka1,a2,...,ak be the complete k k-partite graph of order n = i=1 ai. If at most one value ai is greater than one, then ∼ ∼ γsp(Ka1,a2,...,ak )= n−1 as in such a case Ka1,a2,...,ak = Kn or Ka1,a2,...,ak = Kn−ai +Nai , where G + H denotes the joinP of graphs G and H. As shown in [4], these cases are included in the family of graphs with super domination number equal to n−1. On the other hand, it is not difficult to show that if there are at least two values ai, aj ≥ 2, then γsp(Ka1,a2,...,ak )= n − 2. We leave the details to the reader. In summary, we can state the following. n − 1 if at most one value ai is greater than one. γ (K 1 2 )= sp a ,a ,...,ak n − 2 otherwise. 2 The particular case of paths and cycles was studied in [14]. Theorem 2. [14] For any integer n ≥ 3, n γ (P )= . sp n 2 l m n 2 , n ≡ 0, 3 (mod 4); γ (C )= sp n n+1 , otherwise. 2 It was shown in [4] that the problem of computing γsp(G) is NP-hard. This suggests that finding the super domination number for special classes of graphs or obtaining good bounds on this invariant is worthy of investigation. In particular, the super domination number of lexicographic product graphs and joint graphs was studied in [4] and the case of rooted product graphs was studied in [13]. In this article we study the problem of finding exact values or sharp bounds for the super domination number of graphs. The article is organised as follows. In Section 2, we study the relationships between γsp(G) and several parameters of G, including the number of twin equivalence classes, the domination number, the secure domination number, the matching number, the 2-packing number, the vertex cover number, etc. In Section 3, we obtain a closed formula for the super domination number of any corona graph, while in Section 4 we study the problem of finding the exact values or sharp bounds for the super domination number of Cartesian product graphs and express these in terms of invariants of the factor graphs. 2 Relationship between the super domination number and other parameters of graphs A matching, also called an independent edge set, on a graph G is a set of edges of G such that no two edges share a vertex in common. A largest matching (commonly known as a maximum matching or maximum independent edge set) exists for every graph. The size of this maximum matching is called the matching number and is denoted by α′(G). Theorem 3. For any graph G of order n, ′ γsp(G) ≥ n − α (G). 3 ∗ ∗ Proof. Let D be a γsp(G)-set and let D ⊆ D be a set of cardinality |D | = |D| such that for every u ∈ D there exists u∗ ∈ D∗ such that N(u∗) ∩ D = {u}. Since M = {u∗u ∈ E(G): u∗ ∈ D∗ and u ∈ D} ′ is a matching, we have that n − γsp(G)= |D| = |M| ≤ α (G), as required. As a simple example of a graph where the bound above is achieved we can take ∼ k ′ ′ G = K1 +(∪ K2). In this case α (G)= k, n =2k +1 and γsp(G)= k +1 = n−α (G). n ′ Another example is γsp(Cn)= ⌈ 2 ⌉ = n − α (Cn) whenever n ≡ 0, 3 (mod 4). A vertex cover of G is a set X ⊂ V (G) such that each edge of G is incident to at least one vertex of X. A minimum vertex cover is a vertex cover of smallest possible cardinality. The vertex cover number β(G) is the cardinality of a minimum vertex cover of G. Theorem 4. (König [12], Egerváry [5]) For bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover. We use Theorems 3 and 4 to derive the following result. Theorem 5. For any bipartite graph G of order n, γsp(G) ≥ n − β(G). The bound above is attained, for instance, for any star graph and for any hy- k−1 percube graph. It is well-known that for the hypercube Qk, β(Qk)=2 (see, for k−1 instance, [8]) and in Section 4 we will show that γsp(Qk)=2 . An independent set of G is a set X ⊆ V (G) such that no two vertices in X are adjacent in G, and the independence number of G, α(G), is the cardinality of a largest independent set of G. The following well-known result, due to Gallai, states the relationship between the independence number and the vertex cover number of a graph. Theorem 6. (Gallai’s theorem) For any graph G of order n, α(G)+ β(G)= n. From Theorems 5 and 6 we deduce the following tight bound. Theorem 7. For any bipartite graph G of order n, γsp(G) ≥ α(G). 4 A set S ⊆ V (G) is said to be a secure dominating set of G if it is a dominating set and for every v ∈ S there exists u ∈ N(v) ∩ S such that (S \{u}) ∪{v} is a dominating set. The secure domination number, denoted by γs(G), is the minimum cardinality among all secure dominating sets. This type of domination was introduced by Cockayne et al. in [3]. Theorem 8. For any graph G, γsp(G) ≥ γs(G). ∗ ∗ Proof. Let S ⊆ V (G)bea γsp(G)-set. For each v ∈ S let v ∈ S such that N(v )∩S = {v}. Obviously, S and (S \{v∗}) ∪{v} are dominating sets. Therefore, S is a secure dominating set and so γs(G) ≤|S| = γsp(G). The inequality above is tight. For instance, γs(K1,n−1) = γsp(K1,n−1) = n − 1.

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