The Rank-Width of Edge-Colored Graphs

TL;DR

This paper extends the concept of rank-width to edge-colored graphs, establishing its equivalence to clique-width, characterizing bounded rank-width graphs via vertex-minors, and providing algorithms for their analysis.

Contribution

It introduces two new notions of rank-width for edge-colored graphs, proves their equivalence to clique-width, and develops algorithms for their recognition and analysis.

Findings

F-colored graphs of bounded F-rank-width are characterized by a finite list of forbidden vertex-minors.

A cubic-time algorithm decides if an F-colored graph has F-rank-width at most k.

The framework generalizes existing results for undirected graphs to edge-colored and directed graphs.

Abstract

Clique-width is a complexity measure of directed as well as undirected graphs. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss an extension of the notion of rank-width to edge-colored graphs. A C-colored graph is a graph where the arcs are colored with colors from the set C. There is not a natural notion of rank-width for C-colored graphs. We define two notions of rank-width for them, both based on a coding of C-colored graphs by edge-colored graphs where each edge has exactly one color from a field F and named respectively F-rank-width and F-bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for F-colored graphs and prove that F-colored graphs of bounded F-rank-width are characterised by a finite list of F-colored graphs to exclude as vertex-minors. A cubic-time algorithm to decide whether a F-colored graph has F-rank-width (resp. F-bi-rank-width) at most k, for fixed k, is also given. Graph operations to check MSOL-definable properties on F-colored graphs of bounded rank-width are presented. A specialisation of all these notions to (directed) graphs without edge colors is presented, which shows that our results generalise the ones in undirected graphs.