Dynamic monopolies for interval graphs with bounded thresholds

TL;DR

This paper studies the problem of finding minimal dynamic monopolies in graphs, showing polynomial-time solutions for interval graphs with bounded thresholds and NP-hardness for chordal graphs with unbounded thresholds.

Contribution

It introduces polynomial algorithms for minimum dynamic monopolies in interval graphs with bounded thresholds and proves NP-hardness in chordal graphs with unbounded thresholds.

Findings

Polynomial-time algorithm for interval graphs with bounded thresholds

NP-hardness result for chordal graphs with unbounded thresholds

Differentiates complexity based on graph class and threshold bounds

Abstract

For a graph $G$ and an integer-valued threshold function $\tau$ on its vertex set, a dynamic monopoly is a set of vertices of $G$ such that iteratively adding to it vertices $u$ of $G$ that have at least $\tau(u)$ neighbors in it eventually yields the vertex set of $G$. We show that the problem of finding a dynamic monopoly of minimum order can be solved in polynomial time for interval graphs with bounded threshold functions, but is NP-hard for chordal graphs allowing unbounded threshold functions.