αk- and γk-stable graphs

Abstract

A set I of vertices of a graph G is k-independent if the distance between every two vertices of I is at least k+1. The k-independence number, α k ( G), is the cardinality of a maximum k-independent set of G. A set D of vertices of G is k-dominating if every vertex in V( G)− D is at distance at most k from some vertex in D. The k-domination number, γ k ( G), is the cardinality of a minimum k-dominating set of G. A graph G is α k -stable ( γ k -stable) if α k ( G− e)= α k ( G) ( γ k ( G− e)= γ k ( G)) for every edge e of G. We establish conditions under which a graph is α k - and γ k -stable. In particular, we give constructive characterizations of α k - and γ k -stable trees.