In this article we study the parameterized complexity of problems consisting in finding degree-constrained subgraphs, taking as the parameter the number of vertices of the desired subgraph. Namely, given two positive ...
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In this article we study the parameterized complexity of problems consisting in finding degree-constrained subgraphs, taking as the parameter the number of vertices of the desired subgraph. Namely, given two positive integers d and k, we study the problem of finding a d-regular (induced or not) subgraph with at most k vertices and the problem of finding a subgraph with at most k vertices and of minimum degree at least d. The latter problem is a natural parameterization of the d-girth of a graph (the minimum order of an induced subgraph of minimum degree at least d). We first show that both problems are fixed-parameter intractable in general graphs. More precisely, we prove that the first problem is W[1]-hard using a reduction from MULTI-COLOR Clique. The hardness of the second problem (for the non-induced case) follows from an easy extension of an already known result. We then provide explicit fixed-parametertractable (FPT) algorithms to solve these problems in graphs with bounded local treewidth and graphs with excluded minors, using a dynamic programming approach. Although these problems can be easily defined in first-order logic, hence by the results of Frick and Grohe (2001) [23] are FPT in graphs with bounded local treewidth and graphs with excluded minors, the dependence on k of our algorithms is considerably better than the one following from Frick and Grohe (2001) [23]. (C) 2011 Elsevier B.V. All rights reserved.
We present a new algorithm that can output the rank-decomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs it...
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We present a new algorithm that can output the rank-decomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both of these algorithms are fixed-parametertractable, that is, they run in time O(n(3)) where n is the number of vertices / elements of the input, for each constant value of k and any fixed finite field. The previous best algorithm for construction of a branch-decomposition or a rank-decomposition of optimal width due to Oum and Seymour [J. Combin. Theory Ser. B, 97 (2007), pp. 385 - 393] is not fixed-parametertractable.
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