On the isometric path partition problem

TL;DR

This paper investigates the computational complexity of the isometric path cover and partition problems in graphs, proving NP-completeness for general graphs and deriving exact values for specific grid and network structures.

Contribution

It establishes NP-completeness for the isometric path partition problem and provides exact solutions for grids, tori, and Benes networks, advancing understanding of these problems in structured graphs.

Findings

NP-completeness of the problem on general graphs

Exact isometric path cover numbers for grids and tori

Isometric path cover number for Benes networks

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 $k\geq 3$ are NP-complete on general graphs. Fisher and Fitzpatrick \cite{FiFi01} have shown that the isometric path cover number of $(r\times r)$-dimensional grid is $\lceil 2r/3\rceil$. We show that the isometric path cover (partition) number of $(r\times s)$-dimensional grid is $s$ when $r \geq s(s-1)$. We establish that the isometric path cover (partition) number of $(r\times r)$-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.