Edge-trewidth: Algorithmic and combinatorial properties

TL;DR

This paper introduces the concept of edge treewidth, explores its properties, and establishes its computational complexity, showing it is NP-complete to determine whether a graph's edge treewidth is bounded by a given parameter.

Contribution

It defines edge treewidth as a new graph parameter, studies its properties, introduces a weak topological minor relation, and proves NP-completeness of related decision problems.

Findings

Edge-treewidth is closed under weak topological minors.

Edge-treewidth is parametrically equivalent to maximum degree and treewidth of graph blocks.

Deciding if edge-treewidth ≤ k is NP-complete.

Abstract

We introduce the graph theoretical parameter of edge treewidth. This parameter occurs in a natural way as the tree-like analogue of cutwidth or, alternatively, as an edge-analogue of treewidth. We study the combinatorial properties of edge-treewidth. We first observe that edge-treewidth does not enjoy any closeness properties under the known partial ordering relations on graphs. We introduce a variant of the topological minor relation, namely, the weak topological minor relation and we prove that edge-treewidth is closed under weak topological minors. Based on this new relation we are able to provide universal obstructions for edge-treewidth. The proofs are based on the fact that edge-treewidth of a graph is parametetrically equivalent with the maximum over the treewidth and the maximum degree of the blocks of the graph. We also prove that deciding whether the edge-treewidth of a graph is at most k is an NP-complete problem.