New bounds for contagious sets

We consider the following activation process: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least k active neighbors ( k is identical for all vertices of the graph). Our goal is to find a set of minimum size whose activation will result in the entire graph being activated. Call such a set contagious. We give new upper bounds for the size of contagious sets in terms of the degree sequence of the graph. In particular, we prove that if G = ( V , E ) is an undirected graph then the size of a contagious set is bounded by ∑ v ∈ V min { 1 , k d ( v ) + 1 } (where d ( v ) is the degree of v ). Our proof techniques lead to a new proof for a known theorem regarding induced k -degenerate subgraphs in arbitrary graphs.