Some bounds on the p-domination number in trees

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