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47
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.
Collective tree spanners and routing in ATfree related graphs (Extended Abstract)
 IN GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE, LECTURE NOTES IN COMPUT. SCI. 3353
, 2004
"... In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in ATfree graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any ..."
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Cited by 9 (8 self)
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In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in ATfree graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT(x, y) ≤ dG(x, y) + r. Among other results, we show that ATfree graphs have a system of two collective additive tree 2spanners (whereas there are trapezoid graphs that do not admit any additive tree 2spanner). Furthermore, based on this collection, we derive a compact and efficient routing scheme. Also, any DSPgraph (there exists a dominating shortest path) admits an additive tree 4spanner, a system of two collective additive tree 3spanners and a system of five collective additive tree 2spanners.
Treewidth and Minimum Fillin on DTrapezoid Graphs
, 1998
"... We show that the minimum llin and the minimum interval graph completion of a dtrapezoid graph can be computed in time O(n d). We also show that the treewidth and the pathwidth of a dtrapezoid graph can be computed in time O(n tw(G)^{d1}). In both cases, d is supposed to be a fixed positive integ ..."
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Cited by 9 (3 self)
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We show that the minimum llin and the minimum interval graph completion of a dtrapezoid graph can be computed in time O(n d). We also show that the treewidth and the pathwidth of a dtrapezoid graph can be computed in time O(n tw(G)^{d1}). In both cases, d is supposed to be a fixed positive integer and it is required that a suitable intersection model of the given dtrapezoid graph is part of the input. As a consequence, each of the four graph parameters can be computed in time O(n^2) for trapezoid graphs and thus for permutation graphs even if no intersection model is part of the input.
Interval completion with few edges
 In STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing
, 2007
"... We present an algorithm with runtime O(k 2k n 3 m) for the following NPcomplete problem [8, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the longstanding open question [17, 6, 24, 13], first p ..."
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Cited by 9 (1 self)
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We present an algorithm with runtime O(k 2k n 3 m) for the following NPcomplete problem [8, problem GT35]: Given an arbitrary graph G on n vertices and m edges, can we obtain an interval graph by adding at most k new edges to G? This resolves the longstanding open question [17, 6, 24, 13], first posed by Kaplan, Shamir and Tarjan, of whether this problem could be solved in time f(k) · n O(1). The problem has applications in Physical Mapping of DNA [11] and in Profile Minimization for Sparse Matrix Computations [9, 25]. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope. Our algorithm performs bounded search among possible ways of adding edges to a graph to obtain an interval graph, and combines this with a greedy algorithm when graphs of a certain structure are reached by the search. The presented result is surprising, as it was not believed that a bounded search tree algorithm would suffice to answer the open question affirmatively.
On the Linear Structure and CliqueWidth of Bipartite Permutation Graphs
, 2001
"... Bipartite permutation graphs have several nice characterizations in terms of vertex ordering. Besides, as ATfree graphs, they have a linear structure in the sense that any connected bipartite permutation graph has a dominating path. In the present paper, we elaborate the linear structure of bipa ..."
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Cited by 7 (3 self)
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Bipartite permutation graphs have several nice characterizations in terms of vertex ordering. Besides, as ATfree graphs, they have a linear structure in the sense that any connected bipartite permutation graph has a dominating path. In the present paper, we elaborate the linear structure of bipartite permutation graphs by showing that any connected graph in the class can be stretched into a "path" with "edges" being chain graphs. A particular consequence from the obtained characterization is that the cliquewidth of bipartite permutation graphs is unbounded, which refines a recent result of Golumbic and Rotics for permutation graphs.
Domination and Total Domination on Asteroidal TripleFree Graphs
, 1996
"... We present the first polynomial time algorithms for solving the NPcomplete graph problems DOMINATING SET and TOTAL DOMINATING SET when restricted to asteroidal triplefree graphs. We also present algorithms to compute a minimum cardinality dominating set and a minimum cardinality total dominating s ..."
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Cited by 6 (2 self)
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We present the first polynomial time algorithms for solving the NPcomplete graph problems DOMINATING SET and TOTAL DOMINATING SET when restricted to asteroidal triplefree graphs. We also present algorithms to compute a minimum cardinality dominating set and a minimum cardinality total dominating set on a large superclass of the asteroidal triplefree graphs, called DDPgraphs. These algorithms can be implemented to run in time O(n 6 ) on asteroidaltriple free graphs and in time O(n 7 ) on DDPgraphs. 1 Introduction Asteroidal triplefree graphs (short ATfree graphs) form a large class of graphs containing interval, permutation, trapezoid and cocomparability graphs. Since 1989 ATfree graphs have been studied extensively by Corneil, Olariu and Stewart. They have published a collection of papers presenting many structural and algorithmic properties of ATfree graphs (see e.g. [1013]). By now the knowledge on the algorithmic complexity of NPcomplete graph problems when restri...
Interval Completion is Fixed Parameter Tractable
 IN PROCEEDINGS OF THE 39TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING, STOC 2007
, 2006
"... We give an algorithm with runtime O(k 2k n 3 m) for the NPcomplete problem [GT35 in 6] of deciding whether a graph on n vertices and m edges can be turned into an interval graph by adding at most k edges. We thus prove that this problem is fixed parameter tractable (FPT), settling a longstanding o ..."
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Cited by 6 (2 self)
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We give an algorithm with runtime O(k 2k n 3 m) for the NPcomplete problem [GT35 in 6] of deciding whether a graph on n vertices and m edges can be turned into an interval graph by adding at most k edges. We thus prove that this problem is fixed parameter tractable (FPT), settling a longstanding open problem [13, 5, 19, 11]. The problem has applications in Physical Mapping of DNA [9] and in Profile Minimization for Sparse Matrix Computations [7, 20]. For the first application, our results show tractability for the case of a small number k of false negative errors, and for the second, a small number k of zero elements in the envelope.
Sequential Elimination Graphs
"... A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the order ..."
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Cited by 6 (2 self)
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A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the ordering form a clique. Akcoglu et al [2] define an extension of chordal graphs whereby the neighbors of v that occur later than v in the elimination order have at most k independent vertices. We refer to such graphs as sequentially kindependent graphs. Clearly this extension of chordal graphs also extends the class of (k+1)clawfree graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are sequentially kindependent for small k. In particular, any intersection graph of translates of a convex object in a two dimensional plane is a sequentially 3independent graph; furthermore, any planar graph is a sequentially 3independent graph. For any fixed constant k, we develop simple, polynomial time approximation algorithms for sequentially kindependent graphs with respect to several wellstudied NPcomplete problems based on this ksequentially independent ordering. Our generalized formulation unifies and extends several previously known results. We also consider other classes of sequential elimination graphs.
On the Band, Tree and CliqueWidth of Graphs With Bounded Vertex Degree
, 2002
"... The band, tree and cliquewidth are of primary importance in algorithmic graph theory due to the fact that many problems being NPhard for general graphs can be solved in polynomial time when restricted to graphs where one of these parameters is bounded. It is known that for any fixed Delta >= 3, ..."
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Cited by 5 (2 self)
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The band, tree and cliquewidth are of primary importance in algorithmic graph theory due to the fact that many problems being NPhard for general graphs can be solved in polynomial time when restricted to graphs where one of these parameters is bounded. It is known that for any fixed Delta >= 3, all three parameters are unbounded for graphs with vertex degree at most Delta. In this paper, we distinguish representative subclasses of graphs with bounded vertex degree that have bounded band, tree or cliquewidth. Our proofs are constructive and lead to ecient algorithms for a variety of NPhard graph problems when restricted to those classes.
On the steiner, geodetic and hull numbers of graphs
 Discrete Mathematics
, 2005
"... Given a graph G and a subset W ⊆ V (G), a Steiner Wtree is a tree of minimum order that contains all of W.LetS(W) denote the set of all vertices in G that lie on some Steiner Wtree; we call S(W)theSteiner interval of W.IfS(W)=V (G), then we call W a Steiner set of G. The minimum order of a Steiner ..."
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Cited by 4 (2 self)
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Given a graph G and a subset W ⊆ V (G), a Steiner Wtree is a tree of minimum order that contains all of W.LetS(W) denote the set of all vertices in G that lie on some Steiner Wtree; we call S(W)theSteiner interval of W.IfS(W)=V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G. Given two vertices u, v in G, a shortest u − v path in G is called a u − v geodesic. Let I[u, v] denote the set of all vertices in G lying on some u − v geodesic, and let J[u, v] denote the set of all vertices in G lying on some induced u − v path. Given a set S ⊆ V (G), let I[S] = � u,v∈S I[u, v], and let J[S] =� u,v∈S J[u, v]. We call I[S] the geodetic closure of S and J[S] themonophonic closure of S. IfI[S] =V (G),