On restricted completions of chordal and trivially perfect graphs

TL;DR

This paper studies the complexity of adding edges to graphs close to trivially perfect or chordal graphs, providing a polynomial algorithm for the former and proving NP-completeness for the latter.

Contribution

It introduces a polynomial-time algorithm for edge addition to trivially perfect graphs and proves NP-completeness for similar problems on chordal graphs.

Findings

Polynomial-time algorithm for trivially perfect graph completion.

NP-completeness of chordal graph completion.

Insight into restricted graph completion problems.

Abstract

Let $G$ be a graph having a vertex $v$ such that $H = G - v$ is a trivially perfect graph. We give a polynomial-time algorithm for the problem of deciding whether it is possible to add at most $k$ edges to $G$ to obtain a trivially perfect graph. This is a slight variation of the well-studied {\sc Edge Completion}, also known as {\sc Minimum Fill-In}, problem. We also show that if $H$ is a chordal graph, then the problem of deciding whether it is possible to add at most $k$ edges to $G$ to obtain a chordal graph is \NP-complete.