New ordering methods to construct contagious sets and induced degenerate subgraphs

Randomly ordering vertices in graphs has been useful in deriving upper bounds for the minimal size of contagious sets (where every vertex has threshold k) and lower bounds on the cardinality of k-degenerate induced subgraphs. We introduce a new method of randomly picking permutations from the family of all permutations for which the degree sequence is either nonincreasing or nondecreasing, and we show that this method may result in improved bounds for certain graphs for contagious sets or k-degenerate induced subgraphs. We also reanalyze an algorithm by Reichman for finding contagious sets and obtain a stronger upper bound on the size of the contagious set produced by this algorithm.