Claw-Free Graphs with Equal 2-Domination and Domination Numbers

For a graph G a subsetD of the vertex set of G is a k-dominating set if every vertex not in D has at least k neighbors in D. The k-domination number γk(G) is the minimum cardinality among the k-dominating sets of G. Note that the 1-domination number γ1(G) is the usual domination number γ(G). Fink and Jacobson showed in 1985 that the inequality γk(G) ≥ γ(G) + k − 2 is valid for every connected graph G. In this paper, we concentrate on the case k = 2, where γk can be equal to γ, and we characterize all claw-free graphs and all line graphs G with γ(G) = γ2(G). 1. Terminology and Introduction We consider finite, undirected, and simple graphs G with vertex set V = V(G) and edge set E = E(G). The number of vertices |V(G)| of a graph G is called the order of G and is denoted by n(G). The neighborhood N(v) = NG(v) of a vertex v consists of the vertices adjacent to v and d(v) = dG(v) = |N(v)| is the degree of v. The closed neighborhood of v is the set N[v] = NG[v] = N(v) ∪ {v}. By δ(G) and ∆(G), we denote the minimum degree and the maximum degree of the graph G, respectively. For a subset S ⊆ V, we define by G[S] the subgraph induced by S. If x and y are vertices of a connected graph G, then we denote with dG(x, y) the distance between x and y in G, i.e. the length of a shortest path between x and y. With Kn we denote the complete graph on n vertices and with Cn the cycle of length n. We refer to the complete bipartite graph with partition sets of cardinality p and q as the graph Kp,q. A block is a maximal connected subgraph without cut-vertices. A graph G is a block-cactus graph if every block of G is either a complete graph or a cycle. G is a cactus graph if every block of G is a cycle or a K2. If we substitute each edge in a non-trivial tree by two parallel edges and then subdivide each edge, then we speak of a C4-cactus. Let G and H be two graphs. For a vertex v ∈ V(G), we say that the graph G arises by inflating the vertex v to the graph H if the vertex v is substituted by a set Sv of n(H) new vertices and a set of edges such that G[Sv] H and every vertex in Sv is connected to every neighbor of v in G by an edge. The cartesian product of two graphs G1 and G2 is the graph G1 × G2 with vertex set V(G1) × V(G2) and vertices (u1, u2) and (v1, v2) are adjacent if and only if either u1 = v1 and u2v2 ∈ E(G2) or u2 = v2 and 2010 Mathematics Subject Classification. Primary 05C69