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Decomposition of evenholefree graphs with star cutsets and 2joins
, 2008
"... In this paper we consider the class of simple graphs defined by excluding, as induced subgraphs, even holes (i.e. chordless cycles of even length). These graphs are known as evenholefree graphs. We prove a decomposition theorem for evenholefree graphs, that uses star cutsets and 2joins. This is ..."
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In this paper we consider the class of simple graphs defined by excluding, as induced subgraphs, even holes (i.e. chordless cycles of even length). These graphs are known as evenholefree graphs. We prove a decomposition theorem for evenholefree graphs, that uses star cutsets and 2joins. This is a significant strengthening of the only other previously known decomposition of evenholefree graphs, by Conforti, Cornuéjols, Kapoor and Vušković, that uses 2joins and star, double star and triple star cutsets. It is also analogous to the decomposition of Berge (i.e. perfect) graphs with skew cutsets, 2joins and their complements, by Chudnovsky, Robertson, Seymour and Thomas. The similarity between evenholefree graphs and Berge graphs is higher than the similarity between evenholefree graphs and simply oddholefree graphs, since excluding a 4hole, automatically excludes all antiholes of length at least 6. In a graph that does not contain a 4hole, a skew cutset reduces to a star cutset, and a 2join in the complement implies a star cutset, so in a way it was expected that evenholefree graphs can be decomposed with just the star cutsets and 2joins. A consequence of this decomposition theorem is a recognition algorithm for evenholefree graphs that is significantly faster than the previously known ones. Key words: Evenholefree graphs, star cutsets, 2joins, recognition algorithm, decomposition. 1
Linear Recognition of Almost (Unit) Interval Graphs
, 2014
"... Give a graph class G and a nonnegative integer k, we use G+kv, G+ke, and G−ke to denote the classes of graphs that can be obtained from some graph in G by adding k vertices, adding k edges, and deleting k edges, respectively. They are called almost (unit) interval graphs if G is the class of (unit) ..."
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Give a graph class G and a nonnegative integer k, we use G+kv, G+ke, and G−ke to denote the classes of graphs that can be obtained from some graph in G by adding k vertices, adding k edges, and deleting k edges, respectively. They are called almost (unit) interval graphs if G is the class of (unit) interval graphs. Almost (unit) interval graphs are well motivated from computational biology, where the data ought to be represented by a (unit) interval graph while we can only expect an almost (unit) interval graph for the best. For any fixed k, we give lineartime algorithms for recognizing all these classes, and in the case of membership, our algorithms provide also a specific (unit) interval graph as evidence. When k is part of the input, all the recognition problems are NPcomplete. Our results imply that all of them are fixedparameter tractable parameterized by k, thereby resolving the longstanding open problem on the parameterized complexity of recognizing (unit) interval+ke, first asked by Bodlaender et al. [Comput. Appl. Biosci., 11(1):49–57, 1995]. Moreover, our algorithms for recognizing (unit)interval+kv and (unit)interval−ke have singleexponential dependence on k and linear dependence on the graph size, which significantly improve all previous algorithms for recognizing the same classes. In particular, we show that: (n and m stand for the numbers of vertices and edges respectively in the input graph) • interval−ke can be recognized in time O(6k · (n +m)), improved from O(k2k · n3m) [Heggernes et al., STOC 2007]; • unitinterval−ke can be recognized in time O(4k · (n+m)), improved from O(16k · (m+n)) [Kaplan et al., FOCS 1994]; • interval+kv can be recognized in time O(8k · (n +m)), improved from O(10k · n9) [Cao and Marx,
The kinatree problem for chordal graphs
"... Algorithms for detecting particular induced subgraphs have been the focus of much research recently, mostly related to their connection to many classes of graphs defined by forbidden induced subgraphs. In this context, Chudnovsky and Seymour proposed a useful tool, called threeinatree algorithm w ..."
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Algorithms for detecting particular induced subgraphs have been the focus of much research recently, mostly related to their connection to many classes of graphs defined by forbidden induced subgraphs. In this context, Chudnovsky and Seymour proposed a useful tool, called threeinatree algorithm which solves the following problem in polynomial time: given a graph with three prescribed vertices, test if there is an induced tree containing these vertices. In our paper we deal with a generalization of this problem, known as kinatree problem. For the case where k is part of the input, the problem is known to be NPcomplete. For fixed k, the complexity of this problem is open for k ≥ 4, although there are polynomial time algorithms for restricted cases, such as clawfree graphs and graphs with girth at least k. In this paper we give a O(nm2) time algorithm for the kinatree problem for chordal graphs, even in the case where k is part of the input. Furthermore, the algorithm outputs an induced tree containing the k prescribed vertices when there is such tree.