Not So Easy Problems for Tree Decomposable Graphs

TL;DR

This paper explores the complexity of certain polynomial-time solvable problems on graphs with bounded treewidth, revealing new hardness results and reviewing recent advances in related graph coloring and satisfiability problems.

Contribution

It introduces a new hardness result for minimizing maximum weighted outdegree in bounded treewidth graphs and reviews recent progress on related problems.

Findings

Polynomial-time solvability depends on treewidth bound

New hardness result for maximum weighted outdegree minimization

Review of recent results on coloring and satisfiability problems

Abstract

We consider combinatorial problems that can be solved in polynomial time for graphs of bounded treewidth but where the order of the polynomial that bounds the running time is expected to depend on the treewidth bound. First we review some recent results for problems regarding list and equitable colorings, general factors, and generalized satisfiability. Second we establish a new hardness result for the problem of minimizing the maximum weighted outdegree for orientations of edge-weighted graphs of bounded treewidth.