On Relating Edges in Graphs without Cycles of Length 4

TL;DR

This paper investigates the computational complexity of determining whether an edge in a graph is relating, showing NP-completeness in certain graphs and polynomial-time solvability in others based on cycle length restrictions.

Contribution

It establishes the NP-completeness of the relating edge problem for graphs without cycles of length 4 and 5, and provides a polynomial-time solution for graphs without cycles of length 4 and 6.

Findings

NP-complete for graphs without cycles of length 4 and 5

Polynomial-time algorithm for graphs without cycles of length 4 and 6

Extends understanding of complexity based on cycle length constraints

Abstract

An edge xy is relating in the graph G if there is an independent set S, containing neither x nor y, such that S_{x} and S_{y} are both maximal independent sets in G. It is an NP-complete problem to decide whether an edge is relating (Brown, Nowakowski, Zverovich, 2007). We show that the problem remains NP-complete even for graphs without cycles of length 4 and 5. On the other hand, for graphs without cycles of length 4 and 6, the problem can be solved in polynomial time.