🤖 AI Summary
This paper introduces α-edge-crossing width, a novel graph width parameter defined via the edge-crossing number within bags of a tree-cut decomposition, bridging structural and complexity gaps between tree-partition-width and edge-cut width. Unlike tree-cut width, α-edge-crossing width is incomparable with it but strictly lies between tree-partition-width and edge-cut width. Through rigorous analysis of its hierarchical relationships with these three parameters, the paper establishes the first theoretical foundation for α-edge-crossing width. Methodologically, it develops a fixed-parameter tractable (FPT) algorithm based on dynamic programming and parameterized pruning to decide whether α-edge-crossing width is at most k in time O(2^{O((α+k)log(α+k))} n²). Furthermore, it achieves FPT algorithms for list coloring and precoloring extension when α is fixed. These results significantly extend the applicability of graph width parameters to constraint satisfaction problems.
📝 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 $2alpha^2+5k$ 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.
