K-independence stable graphs upon edge removal

Abstract

Let k be a positive integer and G = (V (G); E(G)) a graph. A subset S of V (G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k 1. The maximum cardinality of a k-independent set of G is the k-independence number k(G). A graph G is called k -stable if k(G e) = k(G) for every edge e of E(G). First we give a necessary and sucien t condition for k -stable graphs. Then we establish four equivalent conditions for k -stable trees.