Resolving prime modules: The structure of pseudo-cographs and galled-tree explainable 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}∈E(G) if and only if the lowest common ancestor lcaT(x,y) of x and y has label “1”. Pseudo-cographs, or, more generally, 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 FGT 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 P4-sparse and P4-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.