Some bounds on the p-domination number in trees Original Research Article

Let p be a positive integer and image a graph. A subset S of V is a p-dominating set if every vertex of image is dominated at least p times, and S is a p-dependent set of G if the subgraph induced by the vertices of S has maximum degree at most image. The minimum cardinality of a p-dominating set a of G is the p-domination number image and the maximum cardinality of a p-dependent set of G is the p-dependence number image. For every positive integer image, we show that for a bipartite graph G, image is bounded above by image, where image is the set of vertices of G of degree at most image, and for every tree T, image is bounded below by image. Moreover, we characterize the trees achieving equality in each bound.