Adding a Tail in Classes of Perfect Graphs

TL;DR

This paper develops linear-time algorithms to compute minimal class completions after adding a tail edge to graphs within certain perfect graph classes, addressing a specific graph augmentation problem.

Contribution

It introduces efficient algorithms for tail addition problems in split, quasi-threshold, threshold, and P4-sparse graphs, leveraging their structural properties.

Findings

Linear-time algorithms for split graphs

Efficient completion methods for quasi-threshold, threshold, and P4-sparse graphs

Structural exploitation enables fast solutions

Abstract

Consider a graph $G$ which belongs to a graph class ${\cal C}$. We are interested in connecting a node $w \not\in V(G)$ to $G$ by a single edge $u w$ where $u \in V(G)$; we call such an edge a \emph{tail}. As the graph resulting from $G$ after the addition of the tail, denoted $G+uw$, need not belong to the class ${\cal C}$, we want to compute a minimum ${\cal C}$-completion of $G+w$, i.e., the minimum number of non-edges (excluding the tail $u w$) to be added to $G+uw$ so that the resulting graph belongs to ${\cal C}$. In this paper, we study this problem for the classes of split, quasi-threshold, threshold, and $P_4$-sparse graphs and we present linear-time algorithms by exploiting the structure of split graphs and the tree representation of quasi-threshold, threshold, and $P_4$-sparse graphs.