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Bidimensional parameters and local treewidth (2004)
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| Venue: | SIAM Journal on Discrete Mathematics |
| Citations: | 31 - 17 self |
BibTeX
@ARTICLE{Demaine04bidimensionalparameters,
author = {Erik D. Demaine and Fedor V. Fomin and Mohammadtaghi Hajiaghayi and Dimitrios M. Thilikos},
title = {Bidimensional parameters and local treewidth},
journal = {SIAM Journal on Discrete Mathematics},
year = {2004},
volume = {18},
pages = {501--511}
}
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Abstract
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, singlecrossing-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 F has the parameter-treewidth property for P if there is a function f(p) such that every graph G ∈ 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 F has the parameter-treewidth property if 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 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. 1
Keyphrases
bidimensional parameter local treewidth parameter-treewidth property minor-closed graph family minor-closed graph class vertex cover graph family planar graph bounded genus main result main tool many domination singlecrossing-minor-free graph feedback vertex graph parameter several fixed-parameter algorithm similar bound sense tight general family large family edge-dominating set new fixed-parameter algorithm several graph theoretic parameter q-dominating set contraction-bidimensional parameter minor-bidimensional parameter
