Article
Feedback vertex set on ATfree graphs
Laboratoire d’Informatique Théorique et Appliquée, Université de Metz, 57045 Metz Cedex 01, France; School of Computing, University of Leeds, Leeds LS2 9JT, UK; Laboratoire d’Informatique Fondamentale d’Orléans (LIFO), Université d’Orléans, BP 6759, 45067 Orléans Cedex 2, France
Discrete Applied Mathematics (Impact Factor: 0.72). 05/2008; DOI: 10.1016/j.dam.2007.10.006 Source: DBLP

Conference Paper: Colouring ATFree graphs
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ABSTRACT: A vertex colouring assigns to each vertex of a graph a colour such that adjacent vertices have different colours. The algorithmic complexity of the Colouring problem, asking for the smallest number of colours needed to vertexcolour a given graph, is known for a large number of graph classes. Notably it is NPcomplete in general, but polynomial time solvable for perfect graphs. A triple of vertices of a graph is called an asteroidal triple if between any two of the vertices there is a path avoiding all neighbours of the third one. Asteroidal triplefree graphs form a graph class with a lot of interesting structural and algorithmic properties. Broersma et al. (ICALP 1997) asked to find out the algorithmic complexity of Colouring on ATfree graphs. Even the algorithmic complexity of the kColouring problem, which asks whether a graph can be coloured with at most a fixed number k of colours, remained unknown for ATfree graphs. First progress was made recently by Stacho who presented an O(n 4 ) time algorithm for 3colouring ATfree graphs (ISAAC 2010). In this paper we show that kColouring on ATfree graphs is in XP, i.e. polynomial time solvable for any fixed k. Even more, we present an algorithm using dynamic programming on an asteroidal decomposition which, for any fixed integers k and a, solves kColouring on any input graph G in time O(f(a,k)·n g(a,k) ), where a denotes the asteroidal number of G, and f(a,k) and g(a,k) are functions that do not depend on n. Hence for any fixed integer k, there is a polynomial time algorithm solving kColouring on graphs of bounded asteroidal number. The algorithm runs in time O(n 8k+2 ) on ATfree graphs.Proceedings of the 20th Annual European conference on Algorithms; 09/2012 
Chapter: Proper Interval Vertex Deletion
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ABSTRACT: Deleting a minimum number of vertices from a graph to obtain a proper interval graph is an NPcomplete problem. At WG 2010 van Bevern et al. gave an O((14k + 14) k + 1 kn 6) time algorithm by combining iterative compression, branching, and a greedy algorithm. We show that there exists a simple greedy O(n + m) time algorithm that solves the Proper Interval Vertex Deletion problem on {claw,net,\allowbreak tent,\allowbreak C4,C5,C6}\{claw,net,\allowbreak tent,\allowbreak C_4,C_5,C_6\}free graphs. Combining this with branching on the forbidden structures claw,net,tent,\allowbreak C4,C5,claw,net,tent,\allowbreak C_4,C_5, and C 6 enables us to get an O(kn 6 6 k ) time algorithm for Proper Interval Vertex Deletion, where k is the number of deleted vertices.12/2010: pages 228238; 
Conference Paper: An exact algorithm for subset feedback vertex set on chordal graphs
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ABSTRACT: Given a graph G=(V,E) and a set S⊆V, a set U⊆V is a subset feedback vertex set of (G,S) if no cycle in G[V∖U] contains a vertex of S. The Subset Feedback Vertex Set problem takes as input G, S, and an integer k, and the question is whether (G,S) has a subset feedback vertex set of cardinality or weight at most k. Both the weighted and the unweighted versions of this problem are NPcomplete on chordal graphs, even on their subclass split graphs. We give an algorithm with running time O(1.6708n) that enumerates all minimal subset feedback vertex sets on chordal graphs with n vertices. As a consequence, Subset Feedback Vertex Set can be solved in time O(1.6708n) on chordal graphs, both in the weighted and in the unweighted case. On arbitrary graphs, the fastest known algorithm for the problems has O(1.8638n) running time.Proceedings of the 7th international conference on Parameterized and Exact Computation; 09/2012
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