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57
Cluster Graph Modification Problems
 DISCRETE APPLIED MATHEMATICS
, 2002
"... In a clustering problem one has to partition a set of elements into homogeneous and wellseparated subsets. From a graph theoretic point of view, a cluster graph is a vertexdisjoint union of cliques. The clustering problem is the task of making fewest changes to the edge set of an input graph so th ..."
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Cited by 68 (5 self)
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In a clustering problem one has to partition a set of elements into homogeneous and wellseparated subsets. From a graph theoretic point of view, a cluster graph is a vertexdisjoint union of cliques. The clustering problem is the task of making fewest changes to the edge set of an input graph so that it becomes a cluster graph. We study the complexity of three variants of the problem. In the Cluster Completion variant edges can only be added. In Cluster Deletion, edges can only be deleted. In Cluster Editing, both edge additions and edge deletions are allowed. We also study these variants when the desired solution must contain a prespecified number of clusters. We show that
Minimal triangulations of graphs: A survey
 DISCRETE MATHEMATICS
"... Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was ..."
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Cited by 38 (3 self)
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Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.
Chordal deletion is fixedparameter tractable
 In 32nd International Workshop on GraphTheoretic Concepts in Computer Science, WG 2006, LNCS Proceedings
, 2004
"... Abstract. It is known to be NPhard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomialtime algorithm for the problem: the running time is f(k) ·n α for some constant α not depending on k and some f depending only on k. For large value ..."
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Cited by 35 (3 self)
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Abstract. It is known to be NPhard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomialtime algorithm for the problem: the running time is f(k) ·n α for some constant α not depending on k and some f depending only on k. For large values of n, such an algorithm is much better than trying all the O(n k) possibilities. Therefore, the chordal deletion problem parameterized by the number k of vertices to be deleted is fixedparameter tractable. This answers an open question of Cai [2]. 1
Parameterized Complexity of Finding Subgraphs with Hereditary Properties
, 2002
"... We consider the parameterized complexity of the following problem under the flamework introduced by Downey and Fellows[4]: Given a graph G, an integer parmneter : and a nontrivial hereditary property H, are there vertices of G that induce a subgraph with property H? This problem has been proved ..."
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Cited by 32 (5 self)
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We consider the parameterized complexity of the following problem under the flamework introduced by Downey and Fellows[4]: Given a graph G, an integer parmneter : and a nontrivial hereditary property H, are there vertices of G that induce a subgraph with property H? This problem has been proved NPhard by Lewis and Yanna kakis[9]. e show that if H includes all independent sets but not all cliques or vice versa, then the problem is hard for the parameterized class kV[1] and is fixed parameter tractable otherwise. In the ibrmer case, if the tbrbidden set of the property is finite, we show, in fact, that the probleln is W[1]complete (see [] for definitions). Our prooil, both of the tractability as well as the hardness ones, involve clever use of Ramsey nmnbers.
Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems
 Algorithmica
, 2004
"... We present a framework for an automated generation of exact search tree algorithms for NPhard problems. The purpose of our approach is twofoldrapid development and improved upper bounds. ..."
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Cited by 30 (10 self)
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We present a framework for an automated generation of exact search tree algorithms for NPhard problems. The purpose of our approach is twofoldrapid development and improved upper bounds.
GraphModeled Data Clustering: FixedParameter Algorithms for Clique Generation
 In Proc. 5th CIAC, volume 2653 of LNCS
, 2003
"... We present e#cient fixedparameter algorithms for the NPcomplete edge modification problems Cluster Editing and Cluster Deletion. ..."
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Cited by 23 (7 self)
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We present e#cient fixedparameter algorithms for the NPcomplete edge modification problems Cluster Editing and Cluster Deletion.
Additive approximation for edgedeletion problems
 Proc. of FOCS 2005
, 2005
"... A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edgedeletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G ..."
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Cited by 19 (8 self)
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A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edgedeletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E ′ P (G). The first result of this paper states that the edgedeletion problem can be efficiently approximated for any monotone property. • For any fixed ɛ> 0 and any monotone property P, there is a deterministic algorithm, which given a graph G = (V, E) of size n, approximates E ′ P (G) in linear time O(V  + E) to within an additive error of ɛn2. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E ′ P. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. 1. If there is a bipartite graph that does not satisfy P, then there is a δ> 0 for which it is
Minimumflip supertrees: complexity and algorithms
 IEEE/ACM Transactions on Computational Biology and Bioinformatics
, 2006
"... Abstract—The input to a supertree problem is a collection of phylogenetic trees that intersect pairwise in their leaf sets; the goal is to construct a single tree that retains as much as possible of the information in the input. This task is complicated by inconsistencies due to errors. We consider ..."
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Cited by 14 (1 self)
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Abstract—The input to a supertree problem is a collection of phylogenetic trees that intersect pairwise in their leaf sets; the goal is to construct a single tree that retains as much as possible of the information in the input. This task is complicated by inconsistencies due to errors. We consider the case where the input trees are rooted and are represented by the clusters they exhibit. The problem is to find the minimum number of flips needed to resolve all inconsistencies, where each flip moves a taxon into or out of a cluster. We prove that the minimumflip problem is NPcomplete, but show that it is fixedparameter tractable and give approximation algorithms for special cases. Index Terms—Phylogenetic tree, supertree, tree assembly, NPcompleteness. Ç 1
Parameterized Complexity of Eulerian Deletion Problems
 ALGORITHMICA
"... We study a family of problems where the goal is to make a graph Eulerian, i.e., connected and with all the vertices having even degrees, by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions ..."
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Cited by 12 (0 self)
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We study a family of problems where the goal is to make a graph Eulerian, i.e., connected and with all the vertices having even degrees, by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc. The collection of results shows an interesting contrast: while the nodedeletion variants remain intractable, i.e., W[1]hard for all the studied cases, edgedeletion problems are either fixedparameter tractable or polynomialtime solvable. Of particular interest is a randomized FPT algorithm for making an undirected graph Eulerian by deleting the minimum number of edges, based on a novel application of the colour coding technique. For versions that remain NPcomplete but fixedparameter tractable we consider also possibilities of polynomial kernelization; unfortunately, we prove that this is not possible unless NP ⊆ coNP/poly.
On the (non)existence of polynomial kernels for Plfree edge modi problems
 In: V. Raman and S. Saurabh (Eds.): IPEC 2010, LNCS 6478:147{157
, 2010
"... Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property Π consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V,E M F) satisfies the property Π. In the Π edgecompletion problem, the set F of edges is constrained t ..."
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Cited by 11 (2 self)
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Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property Π consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V,E M F) satisfies the property Π. In the Π edgecompletion problem, the set F of edges is constrained to be disjoint from E; in the Π edgedeletion problem, F is a subset of E; no constraint is imposed on F in the Π edgeedition problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if Π is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three Π edgemodification problems are FPT [4]. It was then natural to ask [4] whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlström answered this question negatively [15]. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlström asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Plfree and Clfree edgedeletion problems for large enough l. 1