The geodesic cover problem for butterfly networks

TL;DR

This paper determines the minimum number of geodesics needed to cover all vertices and edges of an r-dimensional butterfly network, providing exact formulas for these cover numbers.

Contribution

It establishes exact formulas for the geodesic and edge geodesic cover numbers of r-dimensional butterfly networks, a problem previously only partially solved.

Findings

Geodesic cover number of r-dimensional butterfly is eiling of (2/3) ^r.

Edge geodesic cover number of r-dimensional butterfly is 2^r.

Provides exact solutions for these cover problems in butterfly networks.

Abstract

A geodesic cover, also known as an isometric path cover, of a graph is a set of geodesics which cover the vertex set of the graph. An edge geodesic cover of a graph is a set of geodesics which cover the edge set of the graph. The geodesic (edge) cover number of a graph is the cardinality of a minimum (edge) geodesic cover. The (edge) geodesic cover problem of a graph is to find the (edge) geodesic cover number of the graph. Surprisingly, only partial solutions for these problems are available for most situations. In this paper we demonstrate that the geodesic cover number of the $r$-dimensional butterfly is $\lceil (2/3)2^r\rceil$ and that its edge geodesic cover number is $2^r$.