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}βE(G) if and only if the lowest common
ancestor lcaTβ(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 FGTβ 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 P4β-sparse and
P4β-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.Comment: 18 pages, 3 figure