Algorithms for square-$3PC(\cdot, \cdot)$-free Berge graphs

TL;DR

This paper introduces a combinatorial algorithm with polynomial complexity for finding maximum weight cliques in a new class of Berge graphs characterized by the absence of certain odd holes, antiholes, and specific path configurations.

Contribution

It presents the first polynomial-time algorithm for maximum weight clique detection in the class of square-3PC-free Berge graphs, generalizing previous graph classes.

Findings

Algorithm runs in O(n^7) time.

Successfully detects maximum weight cliques.

Addresses subgraph detection problems in the new class.

Abstract

We consider the class of graphs containing no odd hole, no odd antihole, and no configuration consisting of three paths between two nodes such that any two of the paths induce a hole, and at least two of the paths are of length 2. This class generalizes claw-free Berge graphs and square-free Berge graphs. We give a combinatorial algorithm of complexity $O(n^{7})$ to find a clique of maximum weight in such a graph. We also consider several subgraph-detection problems related to this class.