Elsevier

Discrete Mathematics

Volume 312, Issue 10, 28 May 2012, Pages 1812-1814
Discrete Mathematics

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New bounds for contagious sets

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Abstract

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 vVmin{1,kd(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.

Keywords

Bootstrap percolation
d-degenerate graphs

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Work supported in part by The Israel Science Foundation (grant No. 873/08).