On quasi-planar graphs: Clique-width and logical description

Abstract

Motivated by the construction of FPT graph algorithms parameterized by clique-width or tree-width, we study graph classes for which treewidth and clique-width are linearly related. This is the case for all graph classes of bounded expansion, but in view of concrete applications, we want to have "small" constants in the comparisons between these width parameters.We focus our attention on graphs that can be drawn in the plane with limited edge crossings, for an example, at most p crossings for each edge. These graphs are called p-planar. We consider a more general situation where the graph of edge crossings must belong to a fixed class of graphs D. For p-planar graphs, D is the class of graphs of degree at most p. We prove that tree-width and clique-width are linearly related for graphs drawable with a graph of crossings of bounded average degree.We prove that the class of 1-planar graphs, although conceptually close to that of planar graphs, is not characterized by a monadic second-order sentence. We identify two subclasses that are.

Clique-width for hereditary graph classes

“…Corneil and Rotics [50] improved this bound by showing that cw(G) ≤ 3 • 2 tw(G)−1 for every graph G. They also proved that for every k, there is a graph G with tw(G) = k and cw(G) ≥ 2 tw(G) 2 −1 . The following result shows that for restricted graph classes the two parameters may be equivalent (see [53,54] for graph classes for which treewidth and clique-width are even linearly related). As mentioned in Section 1, boundedness of clique-width has been determined for many hereditary graph classes.…”

Clique-Width for Hereditary Graph Classes

“…The following result shows that for restricted graph classes the two parameters may be equivalent (see [53,54] for graph classes for which treewidth and clique-width are even linearly related).…”

Surveys in Combinatorics 2019

“…Corneil and Rotics [50] improved this bound by showing that cw(G) ≤ 3 • 2 tw(G)−1 for every graph G. They also proved that for every k, there is a graph G with tw(G) = k https://doi.org/10.1017/9781108649094.002 Published online by Cambridge University Press −1 . The following result shows that for restricted graph classes the two parameters may be equivalent (see [53,54] for graph classes for which treewidth and clique-width are even linearly related).…”