Maximum Cardinality Neighbourly Sets in Quadrilateral Free Graphs

TL;DR

This paper presents an efficient O(n^2) algorithm for finding maximum cardinality neighbourly sets in quadrilateral-free graphs, improving significantly over previous algorithms.

Contribution

It introduces a new O(n^2) algorithm for the problem, surpassing earlier polynomial-time solutions with higher complexity.

Findings

The algorithm runs in quadratic time.

It solves the problem efficiently for quadrilateral-free graphs.

It improves upon previous polynomial algorithms.

Abstract

Neighbourly set of a graph is a subset of edges which either share an end point or are joined by an edge of that graph. The maximum cardinality neighbourly set problem is known to be NP-complete for general graphs. Mahdian (M.Mahdian, On the computational complexity of strong edge coloring, Discrete Applied Mathematics, 118:239-248, 2002) proved that it is in polynomial time for quadrilateral-free graphs and proposed an O(n^{11}) algorithm for the same (along with a note that by a straightforward but lengthy argument it can be proved to be solvable in O(n^5) running time). In this paper we propose an O(n^2) time algorithm for finding a maximum cardinality neighbourly set in a quadrilateral-free graph.