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Computational properties of argument systems satisfying graphtheoretic constraints
 Artificial Intelligence
, 2007
"... One difficulty that arises in abstract argument systems is that many natural questions regarding argument acceptability are, in general, computationally intractable having been classified as complete for classes such as NP, coNP, and ¢¡ £. In consequence, a number of researchers have considered me ..."
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Cited by 31 (8 self)
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One difficulty that arises in abstract argument systems is that many natural questions regarding argument acceptability are, in general, computationally intractable having been classified as complete for classes such as NP, coNP, and ¢¡ £. In consequence, a number of researchers have considered methods for specialising the structure of such systems so as to identify classes for which efficient decision processes exist. In this paper the effect of a number of graphtheoretic restrictions is considered: ¤partite systems (¤¦¥¨ § ) in which the set of arguments may be partitioned into ¤ sets each of which is conflictfree; systems in which the numbers of attacks originating from and made upon any argument are bounded; planar systems; and, finally, those of ¤bounded treewidth. For the class of bipartite graphs, it is shown that determining the acceptability status of a specific argument can be accomplished in polynomialtime under both credulous and sceptical semantics. In addition we establish the existence of polynomial time methods for systems having bounded treewidth when deciding the following: whether a given (set of) arguments is credulously accepted; if the system has a nonempty preferred extension; has a stable extension; is coherent;
Pathwidth and ThreeDimensional StraightLine Grid Drawings of Graphs
"... We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for ..."
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Cited by 24 (12 self)
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We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for
Fast robber in planar graphs
, 2008
"... In the cops and robber game, two players play alternately by moving their tokens along the edges of a graph. The first one plays with the cops and the second one with one robber. The cops aim at capturing the robber, while the robber tries to infinitely evade the cops. The main problem consists in m ..."
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Cited by 5 (2 self)
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In the cops and robber game, two players play alternately by moving their tokens along the edges of a graph. The first one plays with the cops and the second one with one robber. The cops aim at capturing the robber, while the robber tries to infinitely evade the cops. The main problem consists in minimizing the number of cops used to capture the robber in a graph. This minimum number is called the copnumber of the graph. If the cops and the robber have the same velocity, g cops are sufficient to capture one robber in any graph with genus g (Schröder, 2001). In the particular case of a grid, 2 cops are sufficient. We investigate the game in which the robber is slightly faster than the cops. In this setting, we prove that the copnumber of planar graphs becomes unbounded. More precisely, we prove that Ω ( √ log n) cops are necessary to capture a fast robber in the n × n squaregrid. This proof consists in designing an elegant evasionstrategy for the robber. Then, it is interesting to ask whether a high value of the copnumber of a planar graph H is related to a large grid G somehow contained in H. We prove that it is not the case when the notion of containment is related to the classical transformations of edge removal, vertex removal, and edge contraction. For instance, we prove that there are graphs with copnumber at most 2 and that are subdivisions of arbitrary large grid. On the positive side, we prove that, if H planar contains a large grid as an induced subgraph, then H has large copnumber. Note that, generally, the copnumber of a graph H is not closed by taking induced subgraphs G, even if H is planar and G is an distancehereditary inducedsubgraph.
Complexity and approximation results for the connected vertex cover problem
"... Abstract. We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APXhard in bipartite graphs and is 5/3approximable in any class of gr ..."
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Cited by 3 (0 self)
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Abstract. We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APXhard in bipartite graphs and is 5/3approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs).
collaborations and interesting discussions. Solving problems alone is boring:).
"... The first person I need to thank is my supervisor Pinar Heggernes. Without her guidance, encouragement and scolding from time to time, this work would not exist. Thank you for taking me as your student, teaching me so much and believing in me from the very start. I have never told you how much this ..."
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The first person I need to thank is my supervisor Pinar Heggernes. Without her guidance, encouragement and scolding from time to time, this work would not exist. Thank you for taking me as your student, teaching me so much and believing in me from the very start. I have never told you how much this meant to me, but I hope this thesis can make up for at least some of it. These three years gave me the opportunity to fulfill many of my dreams, and for this I will always be thankful to you. Another person to whom I owe a lot for his unconditional help, even when he hardly knew me, is Marc Bezem. Your support has been critical in many occasions, including when I had to decide whether to apply for this PhD. Thank you for convincing me to do it, or I would have regretted it forever. I would like to thank also all my coauthors Hans L. Bodlaender, Michael R.