Revolutionaries and spies: Spy-good and spy-bad graphs

TL;DR

This paper analyzes a graph pursuit game involving revolutionaries and spies, establishing bounds and exact values for the minimum number of spies needed in various graph classes and probabilistic settings.

Contribution

It provides new bounds and exact values for the minimal number of spies needed in the revolutionary-spy game across different graph structures and random graph models.

Findings

Lower bound is sharp for certain trees

Bounds related to domination number are established

Exact values are derived for hypercubes and complete multipartite graphs

Abstract

We study a game on a graph $G$ played by $r$ {\it revolutionaries} and $s$ {\it spies}. Initially, revolutionaries and then spies occupy vertices. In each subsequent round, each revolutionary may move to a neighboring vertex or not move, and then each spy has the same option. The revolutionaries win if $m$ of them meet at some vertex having no spy (at the end of a round); the spies win if they can avoid this forever. Let $\sigma(G,m,r)$ denote the minimum number of spies needed to win. To avoid degenerate cases, assume $|V(G)|\ge r-m+1\ge\floor{r/m}\ge 1$. The easy bounds are then $\floor{r/m}\le \sigma(G,m,r)\le r-m+1$. We prove that the lower bound is sharp when $G$ has a rooted spanning tree $T$ such that every edge of $G$ not in $T$ joins two vertices having the same parent in $T$. As a consequence, $\sigma(G,m,r)\le\gamma(G)\floor{r/m}$, where $\gamma(G)$ is the domination number; this bound is nearly sharp when $\gamma(G)\le m$. For the random graph with constant edge-probability $p$, we obtain constants $c$ and $c'$ (depending on $m$ and $p$) such that $\sigma(G,m,r)$ is near the trivial upper bound when $r<c\ln n$ and at most $c'$ times the trivial lower bound when $r>c'\ln n$. For the hypercube $Q_d$ with $d\ge r$, we have $\sigma(G,m,r)=r-m+1$ when $m=2$, and for $m\ge 3$ at least $r-39m$ spies are needed. For complete $k$-partite graphs with partite sets of size at least $2r$, the leading term in $\sigma(G,m,r)$ is approximately $\frac{k}{k-1}\frac{r}{m}$ when $k\ge m$. For $k=2$, we have $\sigma(G,2,r)=\bigl\lceil{\frac{\floor{7r/2}-3}5}\bigr\rceil$ and $\sigma(G,3,r)=\floor{r/2}$, and in general $\frac{3r}{2m}-3\le \sigma(G,m,r)\le\frac{(1+1/\sqrt3)r}{m}$.