On Some Combinatorial Problems in Cographs

TL;DR

This paper explores various combinatorial problems in cographs, providing structural insights and polynomial-time algorithms for problems like vertex separators, connectivity augmentation, and classical optimization tasks.

Contribution

It offers a structural characterization of minimal vertex separators and develops polynomial-time algorithms for several classical problems in cographs.

Findings

Polynomial-time algorithms for listing minimal vertex separators

Efficient solutions for connectivity augmentation in cographs

Dynamic programming framework for classical optimization problems

Abstract

The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper, we study some popular combinatorial problems restricted to cographs. We first present a structural characterization of minimal vertex separators in cographs. Further, we show that listing all minimal vertex separators and the complexity of some constrained vertex separators are polynomial-time solvable in cographs. We propose polynomial-time algorithms for connectivity augmentation problems and its variants in cographs, preserving the cograph property. Finally, using the dynamic programming paradigm, we present a generic framework to solve classical optimization problems such as the longest path, the Steiner path and the minimum leaf spanning tree problems restricted to cographs, our framework yields polynomial-time algorithms for all three problems.