Adding an Edge in a $P_4$-sparse Graph

TL;DR

This paper presents a polynomial-time algorithm for the minimum completion problem in $P_4$-sparse graphs when adding a single edge, by characterizing the structure of optimal solutions.

Contribution

It characterizes the structure of optimal solutions for adding an edge in $P_4$-sparse graphs and provides a polynomial-time algorithm for this problem.

Findings

Optimal solutions have a limited, small set of possible forms.

Polynomial-time algorithm developed for adding an edge in $P_4$-sparse graphs.

Solved specific cases involving spiders and disconnected components.

Abstract

The minimum completion (fill-in) problem is defined as follows: Given a graph family $\mathcal{F}$ (more generally, a property $\Pi$) and a graph $G$, the completion problem asks for the minimum number of non-edges needed to be added to $G$ so that the resulting graph belongs to the graph family $\mathcal{F}$ (or has property $\Pi$). This problem is NP-complete for many subclasses of perfect graphs and polynomial solutions are available only for minimal completion sets. We study the minimum completion problem of a $P_4$-sparse graph $G$ with an added edge. For any optimal solution of the problem, we prove that there is an optimal solution whose form is of one of a small number of possibilities. This along with the solution of the problem when the added edge connects two non-adjacent vertices of a spider or connects two vertices in different connected components of the graph enables us to present a polynomial-time algorithm for the problem.