A note on the Cops & Robber game on graphs embedded in non-orientable surfaces

The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As corollaries, using Schroeder's results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the cop number of graphs embeddable in the Klein Bottle is at most 4, and an upper bound is $3g/2 + 3/2$ for all other $g$.Comment: 5 pages, 1 figur

Adapting Search Theory to Networks

The CSE is interested in the general problem of locating objects in networks. Because of their exposure to search theory, the problem they brought to the workshop was phrased in terms of adapting search theory to networks. Thus, the first step was the introduction of an already existing healthy literature on searching graphs. T. D. Parsons, who was then at Pennsylvania State University, was approached in 1977 by some local spelunkers who asked his aid in optimizing a search for someone lost in a cave in Pennsylvania. Parsons quickly formulated the problem as a search problem in a graph. Subsequent papers led to two divergent problems. One problem dealt with searching under assumptions of fairly extensive information, while the other problem dealt with searching under assumptions of essentially zero information. These two topics are developed in the next two sections

On Necessary and Sufficient Number of Cops in the Game of Cops and Robber in Multidimensional Grids

We theoretically analyze the Cops and Robber Game for the first time in a multidimensional grid. It is shown that for an $n$-dimensional grid, at least $n$ cops are necessary to ensure capture of the robber. We also present a set of cop strategies for which $n$ cops are provably sufficient to catch the robber. Further, for two-dimensional grid, we provide an efficient cop strategy for which the robber is caught even by a single cop under certain conditions.Comment: This is a revised and extended version of the poster paper with the same title that has been presented in the 8th Asian Symposium on Computer Mathematics (ASCM), December 15-17, 2007, Singapor

Revolutionaries and Spies

Let $G = (V,E)$ be a graph and let $r,s,k$ be positive integers. "Revolutionaries and Spies", denoted \cG(G,r,s,k), is the following two-player game. The sets of positions for player 1 and player 2 are $V^r$ and $V^s$ respectively. Each coordinate in $p \in V^r$ gives the location of a "revolutionary" in $G$. Similarly player 2 controls $s$ "spies". We say $u, u' \in V(G)^n$ are adjacent, $u \sim u'$, if for all $1 \leq i \leq n$, $u_i = u'_i$ or ${u_i,u'_i} \in E(G)$. In round 0 player 1 picks $p_0 \in V^r$ and then player 2 picks $q_0 \in V^s$. In each round $i \geq 1$ player 1 moves to $p_i \sim p_{i-1}$ and then player 2 moves to $q_i \sim q_{i-1}$. Player 1 wins the game if he can place $k$ revolutionaries on a vertex $v$ in such a way that player 1 cannot place a spy on $v$ in his following move. Player 2 wins the game if he can prevent this outcome. Let $s(G,r,k)$ be the minimum $s$ such that player 2 can win \cG(G,r,s,k). We show that for $d \geq 2$, $s(\Z^d,r,2)\geq 6 \lfloor \frac{r}{8} \rfloor$. Here $a,b \in \Z^{d}$ with $a \neq b$ are connected by an edge if and only if $|a_i - b_i| \leq 1$ for all $i$ with $1 \leq i \leq d$.Comment: This is the version accepted to appear in Discrete Mathematic

Graphs with Large Girth and Small Cop Number

In this paper we consider the cop number of graphs with no, or few, short cycles. We show that when the girth of $G$ is at least $8$ and the minimum degree is sufficiently large, $\delta \geq (\ln{n})^{\frac{1}{1-\alpha}}$ where $\alpha \in (0,1)$, then $c(G) = o(n \delta^{\beta -\lfloor \frac{g}{4} \rfloor})$ as $\delta \rightarrow \infty$ where $\beta> 1-\alpha$. This extends work of Frankl and implies that if $G$ is large and dense in the sense that $\delta \geq n^{\frac{2}{g} - o(1)}$ while also having girth $g \geq 8$, then $G$ satisfies Meyniel's conjecture, that is $c(G) = O(\sqrt{n})$. Moreover, it implies that if $G$ is large and dense in the sense that there $\delta \geq n^{\epsilon}$ for some $\epsilon >0$, while also having girth $g \geq 8$, then there exists an $\alpha>0$ such that $c(G) = O(n^{1-\alpha})$, thereby satisfying the weak Meyniel's conjecture. Of course, this implies similar results for dense graphs with small, that is $O(n^{1-\alpha})$, numbers of short cycles, as each cycle can be broken by adding a single cop. We also, show that there are graphs $G$ with girth $g$ and minimum degree $\delta$ such that the cop number is at most $o(g (\delta-1)^{(1+o(1))\frac{g}{4}})$. This resolves a recent conjecture by Bradshaw, Hosseini, Mohar, and Stacho, by showing that the constant $\frac{1}{4}$ cannot be improved in the exponent of a lower bound $c(G) \geq \frac{1}{g} (\delta - 1)^{\lfloor \frac{g-1}{4}\rfloor}$.Comment: 7 pages, 0 figures, 0 table