Color-avoiding connected spanning subgraphs with minimum number of edges

TL;DR

This paper studies the maximum edge removal in color-avoiding connected graphs while maintaining their connectivity, proving NP-hardness and providing a polynomial-time approximation algorithm, with extensions to matroids.

Contribution

It establishes NP-hardness of the problem, introduces a polynomial-time approximation algorithm, and analyzes the minimal edges in such graphs, extending to matroid generalizations.

Findings

NP-hardness of the maximum edge removal problem

A polynomial-time approximation algorithm with analyzed performance

Characterization of minimal edges in color-avoiding connected graphs

Abstract

We call a (not necessarily properly) edge-colored graph edge-color-avoiding connected if after the removal of edges of any single color, the graph remains connected. For vertex-colored graphs, similar definitions of color-avoiding connectivity can be given. In this article, we investigate the problem of determining the maximum number of edges that can be removed from a color-avoiding connected graph so that it remains color-avoiding connected. First, we prove that this problem is NP-hard, then we give a polynomial-time approximation algorithm for it. To analyze the approximation factor of this algorithm, we determine the minimum number of edges of color-avoiding connected graphs on a given number of vertices and with a given number of colors. Furthermore, we also consider a generalization of edge-color-avoiding connectivity to matroids.