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47
Approximate Distance Labeling Schemes
, 2000
"... We consider the problem of labeling the nodes of an nnode graph G with short labels in such a way that the distance between any two nodes u; v of G can be approximated eciently (in constant time) by merely inspecting the labels of u and v, without using any other information. We develop such con ..."
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Cited by 46 (18 self)
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We consider the problem of labeling the nodes of an nnode graph G with short labels in such a way that the distance between any two nodes u; v of G can be approximated eciently (in constant time) by merely inspecting the labels of u and v, without using any other information. We develop such constant approximate distance labeling schemes for the classes of trees, bounded treewidth graphs, planar graphs, kchordal graphs, and graphs with a dominating pair (including for instance interval, permutation, and ATfree graphs). We also show lower bounds, and prove that most of our schemes are optimal in length of labels generated and in the quality of the approximation, leaving some open problems.
Complexity classification of some edge modification problems
, 2001
"... In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, c ..."
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Cited by 41 (2 self)
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In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a finite set of forbidden induced subgraphs.
Approximating the Bandwidth for Asteroidal TripleFree Graphs
"... We show that there is an O(n^3) algorithm to approximate the bandwidth of an ATfree graph with worst case performance ratio 2. Alternatively, at the cost of the approximation factor, we can also obtain an O(e + n log n) algorithm to approximate the bandwidth of an ATfree graph within a factor 4 an ..."
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Cited by 35 (1 self)
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We show that there is an O(n^3) algorithm to approximate the bandwidth of an ATfree graph with worst case performance ratio 2. Alternatively, at the cost of the approximation factor, we can also obtain an O(e + n log n) algorithm to approximate the bandwidth of an ATfree graph within a factor 4 and an O(n+ e) algorithm with a factor 6. For the special cases of permutation graphs and trapezoid graphs we obtain O(n log² n) algorithms with worst case performance ratio 2. For cocomparability graphs we obtain an O(n + e) algorithm with worst case performance ratio 3. Finally, we show that there is an O(n² log² n) algorithm to compute the exact bandwidth of chain graphs.
Exact algorithms for treewidth and minimum fillin
 In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004). Lecture Notes in Comput. Sci
, 2004
"... We show that the treewidth and the minimum fillin of an nvertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an nvertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of ATfree g ..."
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Cited by 26 (15 self)
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We show that the treewidth and the minimum fillin of an nvertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an nvertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of ATfree graphs the running time of our algorithms can be reduced to O(1.4142 n).
Linear Time Algorithms for Dominating Pairs in Asteroidal Triplefree Graphs
 SIAM J. Comput
, 1997
"... An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighbourhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is pro ..."
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Cited by 25 (7 self)
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An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighbourhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that ATfree graphs offer a common generalization of interval, permutation, trapezoid, and cocomparability graphs. Previously, the authors have given an existential proof of the fact that every connected ATfree graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected ATfree graphs. The resulting simple algorithm, based on the wellknown Lexicographic BreadthFirst Search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previousl...
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 25 (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. 1 Introduction and
A WideRange Efficient Algorithm For Minimal Triangulation
 Proceedings of SODA'99
, 1999
"... Traditionally, efficient algorithms for computing a minimal triangulation of a graph (i.e. embedding a graph into a triangulated graph by adding an inclusionminimal set of edges) required first computing a special ordering on the vertices of the graph, called a minimal ordering. We give a new algor ..."
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Cited by 20 (9 self)
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Traditionally, efficient algorithms for computing a minimal triangulation of a graph (i.e. embedding a graph into a triangulated graph by adding an inclusionminimal set of edges) required first computing a special ordering on the vertices of the graph, called a minimal ordering. We give a new algorithm which efficiently computes a minimal triangulation using an arbitrary ordering on the vertices. 1 Introduction. Computing a minimal triangulation consists in embedding a given graph into a triangulated graph by adding a set of edges (called a fill). If the set of edges added is inclusionminimal, the fill is said to be minimal, and the corresponding triangulated graph is called a minimal triangulation. Finding a fill that is minimum is NPcomplete ([10]). Given a graph G and any ordering ff on its vertices, an associated fill can be computed by repeatedly choosing the next vertex x in order ff, adding the edges necessary to make the neighborhood of x into a clique (i.e. by making x si...
Additive Tree Spanners
 SIAM JOURNAL ON DISCRETE MATHEMATICS
, 1998
"... A spanning tree of a graph is a kadditive tree spanner whenever the distance of every two vertices in the graph or in the tree differs by at most k. In this paper we show that certain classes of graphs, as distancehereditary graphs, interval graphs, asteroidaltriple free graphs, allow some consta ..."
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Cited by 14 (0 self)
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A spanning tree of a graph is a kadditive tree spanner whenever the distance of every two vertices in the graph or in the tree differs by at most k. In this paper we show that certain classes of graphs, as distancehereditary graphs, interval graphs, asteroidaltriple free graphs, allow some constant k such that every member of the class has some kadditive tree spanner. On the other hand, there are chordal graphs without kadditive tree spanner for arbitrary large k.
Independent Sets In Asteroidal TripleFree Graphs
, 1999
"... An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called ATfree if it does not have an AT. We show that there is an O(n 4 ) ..."
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Cited by 11 (2 self)
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An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called ATfree if it does not have an AT. We show that there is an<F3.502e+05><F3.817e+05><F3.502e+05> O(n<F2.756e+05> 4<F3.817e+05> ) time algorithm to compute the maximum weight of an independent set for ATfree graphs. Furthermore, we obtain<F3.502e+05><F3.817e+05><F3.502e+05> O(n<F2.756e+05> 4<F3.817e+05> ) time algorithms to solve the<F3.728e+05> independent dominating set<F3.817e+05> and the<F3.728e+05> independent perfect dominating set<F3.817e+05> problems on ATfree graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally, we observe that the problems clique and partition into cliques remain NPcomplete when restricted to ATfree graphs.