Edge Forcing in Butterfly Networks

TL;DR

This paper introduces the concept of edge-forcing sets in graphs, proves the NP-completeness of determining their minimum size, and analyzes this parameter specifically for butterfly networks, providing bounds for various dimensions.

Contribution

It defines the edge-forcing set, establishes its computational complexity, and studies its properties in butterfly networks with exact bounds for small dimensions.

Findings

Edge-forcing set concept introduced and formalized.

NP-completeness of the edge-forcing number determination proved.

Exact bounds obtained for butterfly networks of dimensions 2 to 5.

Abstract

A zero forcing set is a set $S$ of vertices of a graph $G$, called forced vertices of $G$, which are able to force the entire graph by applying the following process iteratively: At any particular instance of time, if any forced vertex has a unique unforced neighbor, it forces that neighbor. In this paper, we introduce a variant of zero forcing set that induces independent edges and name it as edge-forcing set. The minimum cardinality of an edge-forcing set is called the edge-forcing number. We prove that the edge-forcing problem of determining the edge-forcing number is NP-complete. Further, we study the edge-forcing number of butterfly networks. We obtain a lower bound on the edge-forcing number of butterfly networks and prove that this bound is tight for butterfly networks of dimensions 2, 3, 4 and 5 and obtain an upper bound for the higher dimensions.