On some combinatorial problems in cographs
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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 finding some constrained vertex separators are linear-time solvable in cographs. We propose polynomial-time algorithms for some 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 and our framework yields polynomial-time algorithms for the three problems.
KeywordsCographs Augmentation problems Vertex separators Hamiltonian path Longest path Steiner path Minimum leaf spanning tree
This work is partially supported by DST-ECRA project - ECR/2017/001442.
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