The pursuit-evasion game on the 1-skeleton graph of a regular polyhedron. I

Abstract

We consider a game between a group of n pursuers and one evader moving with the same maximum velocity along the 1-skeleton graph of a regular polyhedron. The goal of the paper is finding, for each regular polyhedron M, a number N(M) with the following properties: if n ≥ N(M), the group of pursuers wins, while if n < N(M), the evader wins. Part I of the paper is devoted to the case of polyhedra in ℝ³; Part II will be devoted to the case of ℝ^d, d ≥ 5; and Part III, to the case of ℝ⁴.