A short note about pursuit games played on a graph with a given genus

On a finite simple graph G = (X,E), p players pursuers $\text{A}{}_1\text{, ∴, A}{}_{\mathrm{p}}$ and one player evader B who move in turn along the edges of G are considered. The p pursuers $\text{A}{}_1\text{, ∴, A}{}_{\mathrm{p}}$ want to catch B who tries to escape. R. Nowakowski and P. Winkler Discrete Math.43 (1983), 235–240] and A. Quilliot “Thèse de 3° cycle,” pp. 131–145, Université de Paris VI, 1978] already characterized the graphs such that one pursuer is sufficient to catch the evader B. Very recently, M. Aigner and M. Fromme Appl. Discrete Math., in press] proved that if G is a planar graph, three pursuers are sufficient to catch the evader B. This result is extended, showing that if G is a graph with a given genus k, then 3 + 2k pursuers are enough to “arrest” the evader B.