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

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(K5) = α(K3,3) = 3 and α(K5−) = α(K3,3−) = 2, thereby also refining the above result on planar graphs. (K5−(K3,3−) denotes K5(K3,3) minus an edge.) Further, α(Wn) ≤ ⌈n3⌉ + 1, where Wn 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.