1 research outputs found
Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs
The modular decomposition of a graph is a natural construction to capture
key features of in terms of a labeled tree whose vertices are
labeled as "series" (), "parallel" () or "prime". However, full
information of is provided by its modular decomposition tree only,
if is a cograph, i.e., does not contain prime modules. In this case,
explains , i.e., if and only if the lowest common
ancestor of and has label "". Pseudo-cographs,
or, more general, GaTEx graphs are graphs that can be explained by labeled
galled-trees, i.e., labeled networks that are obtained from the modular
decomposition tree of by replacing the prime vertices in 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 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 -sparse and
-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