Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs

TL;DR

This paper characterizes GaTEx graphs, a generalization of cographs, using forbidden subgraphs, and demonstrates their relation to various graph classes, providing linear-time algorithms for key problems.

Contribution

It introduces a new characterization of GaTEx graphs via forbidden subgraphs and links them to many known graph classes, enabling efficient algorithms.

Findings

GaTEx graphs characterized by 25 forbidden induced subgraphs

GaTEx graphs are related to many well-known graph classes

Linear-time algorithms for NP-hard problems on GaTEx graphs

Abstract

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$ is provided by its modular decomposition tree $(T,t)$ only, if $G$ is a cograph, i.e., $G$ does not contain prime modules. In this case, $(T,t)$ explains $G$, i.e., $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_T(x,y)$ of $x$ and $y$ has label "$1$". Pseudo-cographs, or, more general, GaTEx graphs $G$ are graphs that can be explained by labeled galled-trees, i.e., labeled networks $(N,t)$ that are obtained from the modular decomposition tree $(T,t)$ of $G$ by replacing the prime vertices in $T$ by simple labeled cycles. GaTEx graphs can be recognized and labeled galled-trees that explain these graphs can be constructed in linear time. In this contribution, we provide a novel characterization of GaTEx graphs in terms of a set $\mathfrak{F}_{\mathrm{GT}}$ of 25 forbidden induced subgraphs. This characterization, in turn, allows us to show that GaTEx graphs are closely related to many other well-known graph classes such as $P_4$-sparse and $P_4$-reducible graphs, weakly-chordal graphs, perfect graphs with perfect order, comparability and permutation graphs, murky graphs as well as interval graphs, Meyniel graphs or very strongly-perfect and brittle graphs. Moreover, we show that every GaTEx graph as twin-width at most 1 and and provide linear-time algorithms to solve several NP-hard problems (clique, coloring, independent set) on GaTEx graphs by utilizing the structure of the underlying galled-trees they explain.