Dynamic Monopolies for Degree Proportional Thresholds in Connected Graphs of Girth at least Five and Trees

TL;DR

This paper investigates the minimum size of initial vertex sets needed to activate entire graphs under degree-proportional thresholds, providing improved bounds for certain classes of graphs and trees.

Contribution

It establishes new upper bounds on the size of irreversible dynamic monopolies for graphs with girth at least five and for trees, extending previous results with tighter bounds.

Findings

For graphs with girth at least five, $h_{\rho}(G) \leq (2+\epsilon)\rho n(G)$ for small $\rho$.

For trees with order at least $1/\rho$, $h_{\rho}(T) \leq \rho n(T)$ holds.

Improves previous bounds from approximately 5.83 and 4.92 times $\rho n(G)$ to near 2 times $\rho n(G)$ for certain graphs.

Abstract

Let $G$ be a graph, and let $\rho\in (0,1)$. For a set $D$ of vertices of $G$, let the set $H_{\rho}(D)$ arise by starting with the set $D$, and iteratively adding further vertices $u$ to the current set if they have at least $\lceil \rho d_G(u)\rceil$ neighbors in it. If $H_{\rho}(D)$ contains all vertices of $G$, then $D$ is known as an irreversible dynamic monopoly or a perfect target set associated with the threshold function $u\mapsto \lceil \rho d_G(u)\rceil$. Let $h_{\rho}(G)$ be the minimum cardinality of such an irreversible dynamic monopoly. For a connected graph $G$ of maximum degree at least $\frac{1}{\rho}$, Chang (Triggering cascades on undirected connected graphs, Information Processing Letters 111 (2011) 973-978) showed $h_{\rho}(G)\leq 5.83\rho n(G)$, which was improved by Chang and Lyuu (Triggering cascades on strongly connected directed graphs, Theoretical Computer Science 593 (2015) 62-69) to $h_{\rho}(G)\leq 4.92\rho n(G)$. We show that for every $\epsilon>0$, there is some $\rho(\epsilon)>0$ such that $h_{\rho}(G) \leq(2+\epsilon)\rho n(G)$ for every $\rho$ in $(0,\rho(\epsilon))$, and every connected graph $G$ that has maximum degree at least $\frac{1}{\rho}$ and girth at least $5$. Furthermore, we show that $h_{\rho}(T) \leq \rho n(T)$ for every $\rho$ in $(0,1]$, and every tree $T$ that has order at least $\frac{1}{\rho}$.