
Structurally Parameterized dScattered Set
In dScattered Set we are given an (edgeweighted) graph and are asked t...
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Parameterized algorithms and data reduction for the short secluded stpath problem
Given a graph G=(V,E), two vertices s,tβ V, and two integers k,β, we sea...
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Approximate Turing Kernelization for Problems Parameterized by Treewidth
We extend the notion of lossy kernelization, introduced by Lokshtanov et...
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Parameterized algorithms and data reduction for safe convoy routing
We study a problem that models safely routing a convoy through a transpo...
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Parameterized Power Vertex Cover
We study a recently introduced generalization of the Vertex Cover (VC) p...
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Detecting and Enumerating Small Induced Subgraphs in cClosed Graphs
Fox et al. [SIAM J. Comp. 2020] introduced a new parameter, called cclo...
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On structural parameterizations of the selective coloring problem
In the Selective Coloring problem, we are given an integer k, a graph G,...
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FPT and kernelization algorithms for the kinatree problem
The threeinatree problem asks for an induced tree of the input graph containing three mandatory vertices. In 2006, Chudnovsky and Seymour [Combinatorica, 2010] presented the first polynomial time algorithm for this problem, which has become a critical subroutine in many algorithms for detecting induced subgraphs, such as beetles, pyramids, thetas, and even and oddholes. In 2007, Derhy and Picouleau [Discrete Applied Mathematics, 2009] considered the natural generalization to k mandatory vertices, proving that, when k is part of the input, the problem is ππ―complete, and ask what is the complexity of fourinatree. Motivated by this question and the relevance of the original problem, we study the parameterized complexity of kinatree. We begin by showing that the problem is πΆ[1]hard when jointly parameterized by the size of the solution and minimum clique cover and, under the Exponential Time Hypothesis, does not admit an n^o(k) time algorithm. Afterwards, we use Courcelle's Theorem to prove fixedparameter tractability under cliquewidth, which prompts our investigation into which parameterizations admit single exponential algorithms; we show that such algorithms exist for the unrelated parameterizations treewidth, distance to cluster, and distance to cocluster. In terms of kernelization, we present a linear kernel under feedback edge set, and show that no polynomial kernel exists under vertex cover nor distance to clique unless ππ―βπΌπππ―/πππ π. Along with other remarks and previous work, our tractability and kernelization results cover many of the most commonly employed parameters in the graph parameter hierarchy.
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