Between Treewidth and Clique-width
Sigve Hortemo S{\ae}ther, Jan Arne Telle

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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Interconnection Networks and Systems
