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. (Proceedings of the twenty-first annual ACM-SIAM symposium on discrete algorithms, pp 493–502, 2010a; SIAM J Comput 39(5):1941–1956, 2010b). 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 (New width parameters of graphs. The University of Bergen, 2012) 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.
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Notes
In Fomin et al. [9] the problems Graph Coloring, Hamiltonian Cycle and Edge Dominating Set are shown to be FPT only if FPT = W[1], while in Fomin et al. [8] the authors focus on showing that neither of MaxCut, Hamiltonian Cycle and Edge Dominating Set can have a \(f(k)n^{o(k)}\) algorithm parameterized by clique-width unless the exponential Time Hypothethis fails. However, in Fomin et al. [8] they also show that MaxCut is FPT only if FPT = W[1].
A graph class also having this characteristic is the class of complement k-decomposable graphs [1], which combines aspects of co-graphs with bounded treewidth. Interpreted as a parameter, it is incomparable to sm-width, but does lie between treewidth and clique-width. However, we do not see how to compute a decomposition in FPT time.
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Sæther, S.H., Telle, J.A. Between Treewidth and Clique-Width. Algorithmica 75, 218–253 (2016). https://doi.org/10.1007/s00453-015-0033-7
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DOI: https://doi.org/10.1007/s00453-015-0033-7