Abstract
For several graph theoretic parameters such as vertex cover and dominating set, it is known that if their values are bounded by k then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixed-parameter algorithms on minor-closed graph classes such as planar graphs, single-crossing-minor-free graphs, and graphs of bounded genus. In this paper we examine the question whether similar bounds can be obtained for larger minor-closed graph classes, and for general families of parameters including all the parameters where such a behavior has been reported so far.
Given a graph parameter P, we say that a graph family \(\mathcal{F}\) has the parameter-treewidth property for P if there is a function f(p) such that every graph \(G \in \mathcal{F}\) with parameter at most p has treewidth at most f(p). We prove as our main result that, for a large family of parameters called contraction-bidimensional parameters, a minor-closed graph family \(\mathcal{F}\) has the parameter-treewidth property if \(\mathcal{F}\) has bounded local treewidth. We also show “if and only if” for some parameters, and thus this result is in some sense tight. In addition we show that, for a slightly smaller family of parameters called minor-bidimensional parameters, all minor-closed graph families \(\mathcal{F}\) excluding some fixed graphs have the parameter-treewidth property. The bidimensional parameters include many domination and covering parameters such as vertex cover, feedback vertex set, dominating set, edge-dominating set, q-dominating set (for fixed q). We use these theorems to develop new fixed-parameter algorithms in these contexts.
The last author was supported by EC contract IST-1999-14186: Project ALCOM-FT (Algorithms and Complexity) – Future Technologies and by the Spanish CICYT project TIC-2002-04498-C05-03 (TRACER)
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Demaine, E.D., Fomin, F.V., Hajiaghayi, M.T., Thilikos, D.M. (2004). Bidimensional Parameters and Local Treewidth. In: Farach-Colton, M. (eds) LATIN 2004: Theoretical Informatics. LATIN 2004. Lecture Notes in Computer Science, vol 2976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24698-5_15
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DOI: https://doi.org/10.1007/978-3-540-24698-5_15
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