Trees with maximum p p -reinforcement number ?
Let G=(V,E) G = ( V , E ) be a graph and p p a positive integer. The p p -domination number ?p(G) ? p ( G ) is the minimum cardinality of a set D?V D ? V with |NG(x)?D|?p | N G ( x ) ? D | ? p for all x?V?D x ? V ? D . The p p -reinforcement number rp(G) r p ( G ) is the smallest number of edges whose addition to G G results in a graph G? G ? with ?p(G?)<?p(G) ? p ( G ? ) < ? p ( G ) . It is showed by Lu etal. (2013) that rp(T)?p+1 r p ( T ) ? p + 1 for any tree T T and p?2 p ? 2 . This paper characterizes all trees attaining this upper bound when p?3 p ? 3 .
trees p p -domination p p p p p p p -reinforcement number p p p p p