Graffiti.pc on the 2-domination number of a graph

The k-domination number γk(G) of a simple, undirected graph G is the order of a smallest subset D of the vertices of G such that each vertex of G is either in D or adjacent to at least k vertices in D. In 2010, the conjecture-generating computer program, Graffiti.pc, was queried for upperbounds on the 2-domination number. In this paper we prove new upper bounds on the 2-domination number of a graph, some of which generalize to the k-domination number. keywords: 2-domination number, k-domination number, independence number, k-independence number, matching number, core size, path covering number, matching number, neighborhoods. Mathematics Subject Classification: 05C35 Definitions and introduction Let G be a finite simple graph on vertex set V (G) and edge set E(G). A subset D of the vertices of a graph G is a k-dominating set if every vertex of the graph is either in D or adjacent to at least k vertices of D. The k-domination number of a graph G is the order of a smallest k-dominating set, which we denote by γk(G). A subset of the vertices is an independent set if no two vertices in the subset are adjacent. The independence number of graph G, denoted α(G), is the order of a largest independent set. The intersection of all largest independent sets is called the independence-core of the graph and its order the core size, which we denote by αc(G). A subset ∗Work supported in part by the United States Department of Defense and used resources of the Extreme Scale Systems Center at Oak Ridge National Laboratory