HT-graphs have been introduced in [11] and investigated with respect to location problems on grap... more HT-graphs have been introduced in [11] and investigated with respect to location problems on graphs. In this paper two new characterizations of these graphs are given and then it is shown that the central vertex, connected r-domination and Steiner trees problems are linear or almost linear time solvable in HT-graphs.
HT-graphs have been introduced in [11] and investigated with respect to location problems on grap... more HT-graphs have been introduced in [11] and investigated with respect to location problems on graphs. In this paper two new characterizations of these graphs are given and then it is shown that the central vertex, connected r-domination and Steiner trees problems are linear or almost linear time solvable in HT-graphs.
Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms
In case of δ = 0 (trees) and = r = 0, (subtrees of a tree) we recover the result of Alon (2002) a... more In case of δ = 0 (trees) and = r = 0, (subtrees of a tree) we recover the result of Alon (2002) about the transversal and packing numbers of a set-family in which each set is a union of at most κ subtrees of a tree.
In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (lo... more In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. The main contribution of this paper is a new characterization of the hyperbolicity of graphs, via a new parameter which we call rooted insize. This characterization has algorithmic implications in the field of large-scale network analysis. A sharp estimate of graph hyperbolicity is useful, e.g., in embedding an undirected graph into hyperbolic space with minimum distortion [Verbeek and Suri, SoCG'14]. The hyperbolicity of a graph can be computed in polynomialtime, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm (with an additive constant 1) for computing the hyperbolicity of an n-vertex graph G = (V, E) in optimal time O(n 2) (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space. We also show that a similar characterization of hyperbolicity holds for all geodesic metric spaces endowed with a geodesic spanning tree. Along the way, we prove that any complete geodesic metric space (X, d) has such a geodesic spanning tree.
We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. I... more We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. It asks for a given graph to find a shortest path with minimum eccentricity. Although it is NP-hard in general graphs, we demonstrate that a minimum eccentricity shortest path can be found in linear time for distancehereditary graphs (generalizing the previous result for trees) and give a generalised approach which allows to solve the problem in polynomial time for other graph classes. This includes chordal graphs, dually chordal graphs, graphs with bounded tree-length, and graphs with bounded hyperbolicity. Additionally, we give a simple algorithm to compute an additive approximation for graphs with bounded tree-length and graphs with bounded hyperbolicity.
Http Dx Doi Org 10 1080 00207169908804791, Mar 20, 2007
ABSTRACT The k-th power of a graph G is the graph with the same vertex set as G where two vertice... more ABSTRACT The k-th power of a graph G is the graph with the same vertex set as G where two vertices are adjacent iff their distance in G is at most k. We investigate LexBFS-orderings of HHD-free graphsi.e., the graphs where each cycle of length at least five has two chords, with respect to their powers, It is well-known that any LexBFS-ordering of a HHD-free graph is a so-called semisimplicial ordering. We show, that any LexBFS-ordering of a HHD-free graph is a common semisimplicial ordering of all its odd powers. Furthermore, we characterize those HHD-free graphs by forbidden subgraphs for which any LexBFS-ordering of the initial graph is a common perfect elimination ordering of all odd or even powers
Proceedings of the 27th International Workshop on Graph Theoretic Concepts in Computer Science, 2001
ABSTRACT In this paper we show that a very simple and efficient approach can be used to solve the... more ABSTRACT In this paper we show that a very simple and efficient approach can be used to solve the all pairs almost shortest path problem on the class of weakly chordal graphs and its different subclasses. Moreover, this approach works well also on graphs with small size of largest induced cycle and gives a unified way to solve the all pairs almost shortest path and all pairs shortest path problems on different graph classes including chordal, strongly chordal, chordal bipartite, and distance-hereditary graphs.
Proceedings of the 19th International Workshop on Graph Theoretic Concepts in Computer Science, 1993
Recently in several papers, graphs with maximum neighborhood orderings were characterized and tur... more Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By
Proceedings of the Thirteenth Annual Acm Siam Symposium on Discrete Algorithms, 2002
ABSTRACT In this note, we present first linear time algorithms for computing the center and the d... more ABSTRACT In this note, we present first linear time algorithms for computing the center and the diameter of several classes of face regular plane graphs: triangulations with inner vertices of degree ≥ 6, quadrangulations with inner vertices of degree ≥ 4 and the subgraphs of the regular hexagonal grid bounded by a simple circuit of this grid.
We present simple methods for approximating the diameters, radii, and centers of finite sets in δ... more We present simple methods for approximating the diameters, radii, and centers of finite sets in δ-hyperbolic geodesic spaces and graphs. We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δ log 2 n) comparable with that given by M. Gromov.
... ab,hoang-oanh.le}@informatik.uni-rostock.de 2 Department of Computer Science, Kent State Univ... more ... ab,hoang-oanh.le}@informatik.uni-rostock.de 2 Department of Computer Science, Kent State University, Kent, Ohio 44242, USA dragan@cs.kent ... Due to the basic importance of cographs, it is of interest to consider graph classes defined by forbidden one-vertex extensions of a ...
ABSTRACT In this note a greedy algorithm is considered that computes a matching for a graph with ... more ABSTRACT In this note a greedy algorithm is considered that computes a matching for a graph with a given ordering of its vertices, and those graphs are studied for which a vertex ordering exists such that the greedy algorithm always yields maximum cardinality matchings for each induced subgraph. We show that these graphs, called greedy matchable graphs, are a subclass of weakly triangulated graphs and contain strongly chordal graphs and chordal bipartite graphs as proper subclasses. The question when can this ordering be produced efficiently is discussed too.
HT-graphs have been introduced in [11] and investigated with respect to location problems on grap... more HT-graphs have been introduced in [11] and investigated with respect to location problems on graphs. In this paper two new characterizations of these graphs are given and then it is shown that the central vertex, connected r-domination and Steiner trees problems are linear or almost linear time solvable in HT-graphs.
HT-graphs have been introduced in [11] and investigated with respect to location problems on grap... more HT-graphs have been introduced in [11] and investigated with respect to location problems on graphs. In this paper two new characterizations of these graphs are given and then it is shown that the central vertex, connected r-domination and Steiner trees problems are linear or almost linear time solvable in HT-graphs.
Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms
In case of δ = 0 (trees) and = r = 0, (subtrees of a tree) we recover the result of Alon (2002) a... more In case of δ = 0 (trees) and = r = 0, (subtrees of a tree) we recover the result of Alon (2002) about the transversal and packing numbers of a set-family in which each set is a union of at most κ subtrees of a tree.
In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (lo... more In this paper, we study Gromov hyperbolicity and related parameters, that represent how close (locally) a metric space is to a tree from a metric point of view. The study of Gromov hyperbolicity for geodesic metric spaces can be reduced to the study of graph hyperbolicity. The main contribution of this paper is a new characterization of the hyperbolicity of graphs, via a new parameter which we call rooted insize. This characterization has algorithmic implications in the field of large-scale network analysis. A sharp estimate of graph hyperbolicity is useful, e.g., in embedding an undirected graph into hyperbolic space with minimum distortion [Verbeek and Suri, SoCG'14]. The hyperbolicity of a graph can be computed in polynomialtime, however it is unlikely that it can be done in subcubic time. This makes this parameter difficult to compute or to approximate on large graphs. Using our new characterization of graph hyperbolicity, we provide a simple factor 8 approximation algorithm (with an additive constant 1) for computing the hyperbolicity of an n-vertex graph G = (V, E) in optimal time O(n 2) (assuming that the input is the distance matrix of the graph). This algorithm leads to constant factor approximations of other graph-parameters related to hyperbolicity (thinness, slimness, and insize). We also present the first efficient algorithms for exact computation of these parameters. All of our algorithms can be used to approximate the hyperbolicity of a geodesic metric space. We also show that a similar characterization of hyperbolicity holds for all geodesic metric spaces endowed with a geodesic spanning tree. Along the way, we prove that any complete geodesic metric space (X, d) has such a geodesic spanning tree.
We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. I... more We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. It asks for a given graph to find a shortest path with minimum eccentricity. Although it is NP-hard in general graphs, we demonstrate that a minimum eccentricity shortest path can be found in linear time for distancehereditary graphs (generalizing the previous result for trees) and give a generalised approach which allows to solve the problem in polynomial time for other graph classes. This includes chordal graphs, dually chordal graphs, graphs with bounded tree-length, and graphs with bounded hyperbolicity. Additionally, we give a simple algorithm to compute an additive approximation for graphs with bounded tree-length and graphs with bounded hyperbolicity.
Http Dx Doi Org 10 1080 00207169908804791, Mar 20, 2007
ABSTRACT The k-th power of a graph G is the graph with the same vertex set as G where two vertice... more ABSTRACT The k-th power of a graph G is the graph with the same vertex set as G where two vertices are adjacent iff their distance in G is at most k. We investigate LexBFS-orderings of HHD-free graphsi.e., the graphs where each cycle of length at least five has two chords, with respect to their powers, It is well-known that any LexBFS-ordering of a HHD-free graph is a so-called semisimplicial ordering. We show, that any LexBFS-ordering of a HHD-free graph is a common semisimplicial ordering of all its odd powers. Furthermore, we characterize those HHD-free graphs by forbidden subgraphs for which any LexBFS-ordering of the initial graph is a common perfect elimination ordering of all odd or even powers
Proceedings of the 27th International Workshop on Graph Theoretic Concepts in Computer Science, 2001
ABSTRACT In this paper we show that a very simple and efficient approach can be used to solve the... more ABSTRACT In this paper we show that a very simple and efficient approach can be used to solve the all pairs almost shortest path problem on the class of weakly chordal graphs and its different subclasses. Moreover, this approach works well also on graphs with small size of largest induced cycle and gives a unified way to solve the all pairs almost shortest path and all pairs shortest path problems on different graph classes including chordal, strongly chordal, chordal bipartite, and distance-hereditary graphs.
Proceedings of the 19th International Workshop on Graph Theoretic Concepts in Computer Science, 1993
Recently in several papers, graphs with maximum neighborhood orderings were characterized and tur... more Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By
Proceedings of the Thirteenth Annual Acm Siam Symposium on Discrete Algorithms, 2002
ABSTRACT In this note, we present first linear time algorithms for computing the center and the d... more ABSTRACT In this note, we present first linear time algorithms for computing the center and the diameter of several classes of face regular plane graphs: triangulations with inner vertices of degree ≥ 6, quadrangulations with inner vertices of degree ≥ 4 and the subgraphs of the regular hexagonal grid bounded by a simple circuit of this grid.
We present simple methods for approximating the diameters, radii, and centers of finite sets in δ... more We present simple methods for approximating the diameters, radii, and centers of finite sets in δ-hyperbolic geodesic spaces and graphs. We also provide a simple construction of distance approximating trees of δ-hyperbolic graphs G on n vertices with an additive error O(δ log 2 n) comparable with that given by M. Gromov.
... ab,hoang-oanh.le}@informatik.uni-rostock.de 2 Department of Computer Science, Kent State Univ... more ... ab,hoang-oanh.le}@informatik.uni-rostock.de 2 Department of Computer Science, Kent State University, Kent, Ohio 44242, USA dragan@cs.kent ... Due to the basic importance of cographs, it is of interest to consider graph classes defined by forbidden one-vertex extensions of a ...
ABSTRACT In this note a greedy algorithm is considered that computes a matching for a graph with ... more ABSTRACT In this note a greedy algorithm is considered that computes a matching for a graph with a given ordering of its vertices, and those graphs are studied for which a vertex ordering exists such that the greedy algorithm always yields maximum cardinality matchings for each induced subgraph. We show that these graphs, called greedy matchable graphs, are a subclass of weakly triangulated graphs and contain strongly chordal graphs and chordal bipartite graphs as proper subclasses. The question when can this ordering be produced efficiently is discussed too.
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Papers by F. Dragan