On the largest dynamic monopolies of graphs with a given average threshold

TL;DR

This paper studies the maximum size of dynamic monopolies in graphs with fixed average thresholds, exploring bounds, computational complexity, and special cases like forests and planar graphs.

Contribution

It introduces the concept of $Ldyn_t(G)$ as the worst-case dynamic monopoly size for given average thresholds and analyzes its bounds and computational complexity.

Findings

$Ldyn_t(G)$ is a continuous set of integers for fixed average thresholds.

Determines conditions for upper bounds of $Ldyn_t(G)$ proportional to graph size.

Shows $Ldyn_t(G)$ is coNP-hard for planar graphs but polynomial for forests.

Abstract

Let $G$ be a graph and $\tau$ be an assignment of nonnegative integer thresholds to the vertices of $G$. A subset of vertices $D$ is said to be a $\tau$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$ and for any $i\in \{0, \ldots, k-1\}$, each vertex $v$ in $D_{i+1}$ has at least $\tau(v)$ neighbors in $D_0\cup \ldots \cup D_i$. Denote the size of smallest $\tau$-dynamic monopoly by $dyn_{\tau}(G)$ and the average of thresholds in $\tau$ by $\overline{\tau}$. We show that the values of $dyn_{\tau}(G)$ over all assignments $\tau$ with the same average threshold is a continuous set of integers. For any positive number $t$, denote the maximum $dyn_{\tau}(G)$ taken over all threshold assignments $\tau$ with $\overline{\tau}\leq t$, by $Ldyn_t(G)$. In fact, $Ldyn_t(G)$ shows the worst-case value of a dynamic monopoly when the average threshold is a given number $t$. We investigate under what conditions on $t$, there exists an upper bound for $Ldyn_{t}(G)$ of the form $c|G|$, where $c<1$. Next, we show that $Ldyn_t(G)$ is coNP-hard for planar graphs but has polynomial-time solution for forests.