Revolutionaries and spies: Spy-good and spy-bad 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 $\frac{3r}{2m}-3\le \sigma(G,m,r)\le\frac{(1+1/\sqrt3)r}{m}$.Comment: 34 pages, 2 figures. The most important changes in this revision are improvements of the results on hypercubes and random graphs. The proof of the previous hypercube result has been deleted, but the statement remains because it is stronger for m<52. In the random graph section we added a spy-strategy resul