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arXiv · 2110.07706

On the proper interval completion problem within some chordal subclasses

François Dross·Claire Hilaire·Ivo Koch·Valeria Leoni·Nina Pardal·María Inés Lopez Pujato·Vinicius Fernandes dos Santos
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Abstract

Given a property (graph class) $\Pi$, a graph $G$, and an integer $k$, the \emph{$\Pi$-completion} problem consists in deciding whether we can turn $G$ into a graph with the property $\Pi$ by adding at most $k$ edges to $G$. The $\Pi$-completion problem is known to be NP-hard for general graphs when $\Pi$ is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem %when $\Pi$ is the class of proper interval graphs (PIG) within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.

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François Dross, Claire Hilaire, Ivo Koch, Valeria Leoni, Nina Pardal, María Inés Lopez Pujato, Vinicius Fernandes dos Santos. 2021-10-14. On the proper interval completion problem within some chordal subclasses. https://arxiv.org/abs/2110.07706

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