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New ordering methods to construct contagious sets and induced degenerate subgraphs

Connor C. Anderson, Akshaj Balasubramanian, Henry Poskanzer, Daniel Reichman and Gábor N. Sárközy

Vol. 16 (2023), No. 1, 59–68
Abstract

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.

Keywords
contagious sets, degenerate subgraphs
Mathematical Subject Classification
Primary: 05C35
Secondary: 05C69
Milestones
Received: 28 July 2021
Revised: 5 April 2022
Accepted: 20 April 2022
Published: 14 April 2023

Communicated by Kenneth S. Berenhaut
Authors
Connor C. Anderson
Computer Science Department
Worcester Polytechnic Institute
Worcester, MA
United States
Akshaj Balasubramanian
Computer Science Department
Worcester Polytechnic Institute
Worcester, MA
United States
Henry Poskanzer
Computer Science Department
Worcester Polytechnic Institute
Worcester, MA
United States
Daniel Reichman
Computer Science Department
Worcester Polytechnic Institute
Worcester, MA
United States
Gábor N. Sárközy
Computer Science Department
Worcester Polytechnic Institute
Worcester, MA
United States