On a pursuit game played on graphs for which a minor is excluded

Abstract

In a previous paper, Aigner and Fromme ( Discrete Appl. Math. 8 (1984), 1–12) considered a game played on a finite connected graph G where s pursuers try to catch one evader. They introduced c( G) as the minimal number s of pursuers that are sufficient to catch the evader and showed that, in general, c( G) can be arbitrarily high. On the other hand, they proved that c( G) ≤ 3 if G is planar. The present paper relates c( G) to the “forbidden minor concept.” Suppose that the graph H is not a minor of G and that, for a vertex h ϵ V( H) H − h has no isolated vertices. It is shown that this implies c( G) ≤ | E( H − h)|. As a consequence, one finds that, for each graph H, there exists a minimal positive integer α( H) such that c( G) ≤ α( H) when H is not a minor of G and, in addition, α( H) < | E( H)| for each connected H with at least two edges. These results are refined by proving that α( K 5) = α( K 3,3) = 3 and α( K 5 −) = α( K 3,3 −) = 2, thereby also refining the above result on planar graphs. ( K 5 −( K 3,3 −) denotes K 5( K 3,3) minus an edge.) Further, α(W n) ≤ ⌈ n 3 ⌉ + 1 , where W n is the wheel with n rim vertices ( n ≥ 3). In addition, we also establish an upper bound for c( G) in terms of the cross-cap number of G, thus providing a (partial) analogue to a result of Quilliot on the genus. Finally, the relationship between c( G) and simplicial decompositions is studied and a list of open problems is presented.