A new width parameter of graphs based on edge cuts: $\alpha$-edge-crossing width

TL;DR

This paper introduces a new graph width parameter called $oldsymbol{oldsymbol{ ext{ extalpha}-edge-crossing width}}$, explores its relationships with existing parameters, and provides algorithms for computing it, enabling fixed-parameter tractable solutions for certain coloring problems.

Contribution

The paper defines the novel $ ext{ extalpha}-edge-crossing width$ parameter, analyzes its relation to existing parameters, and develops an FPT algorithm for computing it, closing complexity gaps for coloring problems.

Findings

$ ext{ extalpha}-edge-crossing width$ is a new parameter between tree-partition-width and edge-cut width.

The algorithm can determine bounds on $ ext{ extalpha}-edge-crossing width$ in $2^{O(( ext{ extalpha}+k) ext{ extlog}( ext{ extalpha}+k))}n^2$ time.

FPT algorithms for List Coloring and Precoloring Extension are obtained parameterized by $ ext{ extalpha}-edge-crossing width$.

Abstract

We introduce graph width parameters, called $\alpha$-edge-crossing width and edge-crossing width. These are defined in terms of the number of edges crossing a bag of a tree-cut decomposition. They are motivated by edge-cut width, recently introduced by Brand et al. (WG 2022). We show that edge-crossing width is equivalent to the known parameter tree-partition-width. On the other hand, $\alpha$-edge-crossing width is a new parameter; tree-cut width and $\alpha$-edge-crossing width are incomparable, and they both lie between tree-partition-width and edge-cut width. We provide an algorithm that, for a given $n$-vertex graph $G$ and integers $k$ and $\alpha$, in time $2^{O((\alpha+k)\log (\alpha+k))}n^2$ either outputs a tree-cut decomposition certifying that the $\alpha$-edge-crossing width of $G$ is at most $2\alpha^2+5k$ or confirms that the $\alpha$-edge-crossing width of $G$ is more than $k$. As applications, for every fixed $\alpha$, we obtain FPT algorithms for the List Coloring and Precoloring Extension problems parameterized by $\alpha$-edge-crossing width. They were known to be W[1]-hard parameterized by tree-partition-width, and FPT parameterized by edge-cut width, and we close the complexity gap between these two parameters.