On (1,2)-domination in cubic graphs
Introduction
A total dominating set of a graph G with minimum degree at least 1 is a set S of vertices such that every vertex in G has a neighbor in S, where a neighbor of a vertex v in G is a vertex u that is adjacent to v. The total domination number, , of G is the minimum cardinality of a total dominating set of G. For recent books on domination and total domination in graphs, we refer the reader to [7], [8], [12].
In this paper we study a stronger form of total domination. For an integer, a -dominating set in a graph G with minimum degree at least k is a set S of vertices of G such that every vertex in has at least one neighbor in S and every vertex in S has at least k neighbors in S. The -domination number, , of G is the minimum cardinality of a -dominating set of G. We call a -dominating set in G of cardinality a -set of G.
More generally for integers and , an -dominating set of a graph G with minimum degree at least s is a set S of vertices such that every vertex in has at least r neighbors in S and every vertex in S has at least s neighbors in S. The -domination number, , of G is the minimum cardinality of an -dominating set of G. The concept of -domination in graphs was introduced by Cockayne [5] under a more general framework for domination in graphs, and by Favaron, Henning, Puech and Rautenbach [6]. We refer the reader to the PhD thesis by Roux [13] on -domination in graphs, and to the excellent survey by Chellali, Favaron, Hansberg, and Volkmann [3] on k-domination and k-independence in graphs.
For notation and graph theory terminology, we in general follow [12]. Specifically, let G be a graph with vertex set and edge set , and of order and size . The open neighborhood of a vertex v in G, denoted , is the set of neighbors of v, and so . The closed neighborhood of v is . If the graph G is clear from the context, we omit the subscript and write and rather than and , respectively. A cubic graph (also called a 3-regular graph) is a graph in which every vertex has degree 3. For a set of vertices , the subgraph induced by S is denoted by . We denote the subgraph obtained from G by deleting all vertices in S and all edges incident with vertices in S by . The distance between two vertices u and v in a connected graph G, denoted by , is the length of a shortest -path in G.
A path on n vertices is denoted by . We denote by the complete graph on n vertices. The complete bipartite graph with one partite set of size n and the other of size m is denoted by . A complete graph we call a triangle, while a complete graph on four vertices minus one edge we call a diamond. A 2-factor of a graph G is a spanning 2-regular subgraph of G; that is, a 2-factor of G is a collection of vertex-disjoint cycles that contain all vertices of G. We use the standard notation .
Section snippets
Main result
A -dominating set is precisely a total dominating set, and so if G is a graph with minimum degree at least 1, then . A -dominating set is a double total dominating set (also called a 2-tuple total dominating set in the literature) studied, for example, in [11]. The following results are known. We remark that the 2004 result in Theorem 1(a) by Archdeacon et al. [2] was a major breakthrough at the time.
Theorem 1 If G is a cubic graph of order n, then the following holds. (a) ([2], [4]
Proof of Theorem 2
In this section, we present a proof of our main result, namely Theorem 2. Let be a cubic graph with vertex set V and edge set E of order n. We proceed by induction on the order . If , then , and . If , then either G is the prism shown in Fig. 1(a) or G is the complete bipartite graph shown in Fig. 1(b). If , then , while if , then , where the darkened vertices in Fig. 1(a) and 1(b) are examples of a -set of G. Thus,
Sharpness of Theorem 2
In this section, we show the sharpness of Theorem 2. Adopting the terminology and notation of a diamond-necklace coined by Henning and Löwenstein in [9], for an integer, let be the connected cubic graph constructed as follows. Take k disjoint copies of a diamond, where and where is the missing edge in . Let be obtained from the disjoint union of these k diamonds by adding the edges and adding the edge . For notational
Concluding remarks and open problems
Let be the family of connected cubic graphs constructed as follows. For an arbitrary fixed integer, let and be vertex disjoint 2-regular graphs (not necessarily connected) on k vertices. Thus, is either a cycle or a disjoint union of cycles the sum of whose orders is k for . Let G be the graph of order obtained from the disjoint union by adding a perfect matching between vertices of and . Let be the family of all graphs thus constructed which
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Research supported in part by the University of Johannesburg.