On the isometric path partition problem

Abstract

The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric k-path partition problem for k ≥ 3 are NP-complete on general graphs. Fisher and Fitzpatrick [7] have shown that the isometric path cover number of (r × r)-dimensional grid is 2r/3 . We show that the isometric path cover (partition) number of (r × s)-dimensional grid is s when r ≥ s(s − 1). We establish that the isometric path cover (partition) number of (r × r)-dimensional torus is r when r is even and is either r or r + 1 when r is odd. Then, we demonstrate that the isometric path cover (partition) number of an r-dimensional Benes network is 2 r . In addition, we provide partial solutions for the isometric path cover (partition) problems for cylinder and multi-dimensional grids.

The Edge General Position Problem

“…Since gcover e (G) ≥ m(G)/diam(G) (cf. [14]), diam(G) = 2r, and m(G) = 2 • 2r • 2r, we infer that gcover e (G) ≥ 4r.…”