Between Treewidth and Clique-width

TL;DR

This paper introduces sm-width, a new graph parameter between treewidth and clique-width, enabling fixed-parameter tractability for several hard problems on broader graph classes.

Contribution

The paper defines sm-width based on splits and branch decompositions, and proves that key problems are FPT when parameterized by sm-width, broadening tractable cases.

Findings

sm-width is between treewidth and clique-width

Certain unbounded treewidth classes have bounded sm-width

Key problems are FPT with sm-width parameterization

Abstract

Many hard graph problems can be solved efficiently when restricted to graphs of bounded treewidth, and more generally to graphs of bounded clique-width. But there is a price to be paid for this generality, exemplified by the four problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that are all FPT parameterized by treewidth but none of which can be FPT parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7, 8]. We therefore seek a structural graph parameter that shares some of the generality of clique-width without paying this price. Based on splits, branch decompositions and the work of Vatshelle [18] on Maximum Matching-width, we consider the graph parameter sm-width which lies between treewidth and clique-width. Some graph classes of unbounded treewidth, like distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized by sm-width.