文摘
Let p be a positive integer and G=(V,E) a graph. A subset S of V is a p-dominating set if every vertex of V-S 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 p-1. The minimum cardinality of a p-dominating set a of G is the p-domination number γp(G) and the maximum cardinality of a p-dependent set of G is the p-dependence number βp(G). For every positive integer p2, we show that for a bipartite graph G, γp(G) is bounded above by (V+Yp)/2, where Yp is the set of vertices of G of degree at most p-1, and for every tree T, γp(T) is bounded below by βp-1(T). Moreover, we characterize the trees achieving equality in each bound.