Some results on Kr-covered graphs - Favaron O.; Li H.; Plummer M.D. (Utilitas Mathematica, 1998)
A graph G is Kr-covered if every vertex of G is contained in a Kr, i.e., in a complete subgraph on r vertices. Two vertices of a graph G are said to be Kr-adjacent if there is a Kr in G containing them both. A set of vertices D in a Kr-covered graph G is a Kr-dominating set if every vertex of G either belongs to D or is Kr-adjacent to a vertex of D. The Kr-domination number γKr(G) is the size of any smallest Kr-dominating set in G. In this paper, we study properties of Kr-covered graphs. In particular, we obtain a new and much shorter proof of an upper bound first obtained by Henning and Swart [HS] for the K4-domination number of a K4-covered graph and obtain a new bound for the K5-domination number of a K5-covered graph in which the maximum degree does not exceed 7.