Nancy E. Clarke

Characterizations of k-copwin graphs

We give two characterizations of the graphs on which k cops have a winning strategy in the game of Cops and Robber. One of these is in terms of an order relation, and one is in terms of a vertex ordering. Both generalize characterizations known for the case k=1.

Tandem-win graphs

In this version of the Cops and Robber game, the cops move in tandems, or pairs, such that they are at distance at most one after every move. We present a recognition theorem for tandem-win graphs, and a characterization of triangle-free tandem-win graphs.

A Note on the Cops and Robber Game on Graphs Embedded in Non-Orientable Surfaces

We consider the two-player, complete information game of Cops and Robber played on undirected, finite, reflexive graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. Let c(g) be the supremum over all cop numbers of graphs embeddable in a closed orientable surface of genus g, and likewise

A simple method of computing the catch time

{\tilde c(g)}

A note on the Grundy number and graph products

for non-orientable surfaces. It is known (Andreae, 1986) that, for a fixed surface, the maximum over all cop numbers of graphs embeddable in this surface is finite. More precisely, Quilliot (1985) showed that c(g) ≤ 2g + 3, and Schröder (2001) sharpened this to

Constrained cops and robber.

{c(g)\le \frac32g + 3}