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33
Metric graph theory and geometry: a survey
 CONTEMPORARY MATHEMATICS
"... The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of general ..."
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Cited by 44 (14 self)
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The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fibercomplemented graphs, or l1graphs. Several kinds of l1graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or treelike graphs such as distancehereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the
Distance Approximating Trees for Chordal and Dually Chordal Graphs
, 1999
"... In this paper we show that, for each chordal graph G, there is a tree T such that T is a spanning tree of the square G² of G and, for every two vertices, the distance between them in T is not larger than the distance in G plus 2. Moreover, we prove that, if G is a strongly chordal graph or even a ..."
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Cited by 31 (18 self)
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In this paper we show that, for each chordal graph G, there is a tree T such that T is a spanning tree of the square G² of G and, for every two vertices, the distance between them in T is not larger than the distance in G plus 2. Moreover, we prove that, if G is a strongly chordal graph or even a dually chordal graph, then there exists a spanning tree T of G that is an additive 3spanner as well as a multiplicative 4spanner of G. In all cases the tree T can be computed in linear time
Chordal Graphs and Their Clique Graphs
 IN WG ’95
, 1995
"... In the first part of this paper, a new structure for chordal graph is introduced, namely the clique graph. This structure is shown to be optimal with regard to the set of clique trees. The greedy aspect of the recognition algorithms of chordal graphs is studied. A new greedy algorithm that generali ..."
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Cited by 20 (7 self)
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In the first part of this paper, a new structure for chordal graph is introduced, namely the clique graph. This structure is shown to be optimal with regard to the set of clique trees. The greedy aspect of the recognition algorithms of chordal graphs is studied. A new greedy algorithm that generalizes both Maximal cardinality Search (MCS) and Lexicographic Breadth first search is presented. The trace of an execution of MCS is defined and used in two linear time and space algorithms: one builds a clique tree of a chordal graph and the other is a simple recognition procedure of chordal graphs.
Algorithmic aspects of domination in graphs
 in: Handbook of Combinatorial Optimization (D.Z. Du and P. M. Pardalos eds
, 1998
"... Graph theory was founded by Euler [78] in 1736 as a generalization to the solution of the famous problem of the Könisberg bridges. From 1736 to 1936, the same concept as graph, but under different names, was used in various scientific fields as models of real world problems, see the historic book by ..."
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Cited by 10 (3 self)
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Graph theory was founded by Euler [78] in 1736 as a generalization to the solution of the famous problem of the Könisberg bridges. From 1736 to 1936, the same concept as graph, but under different names, was used in various scientific fields as models of real world problems, see the historic book by Biggs, Lloyd and Wilson [19]. This chapter intents to survey the domination problem in graph theory, which is a natural model for many location problems in operations research, from an algorithmic point of view. 1
Clique rDomination and Clique rPacking Problems on Dually Chordal Graphs
, 1997
"... Let be a family of cliques of a graph G =(V,E). Suppose that each clique C of is associated with an integer r(C), where r(C) 0. A vertex vrdominates a clique C of G if d(v, x) r(C) for all x C, where d(v, x) is the standard graph distance. A subset D V is a clique rdominating set ..."
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Cited by 9 (1 self)
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Let be a family of cliques of a graph G =(V,E). Suppose that each clique C of is associated with an integer r(C), where r(C) 0. A vertex vrdominates a clique C of G if d(v, x) r(C) for all x C, where d(v, x) is the standard graph distance. A subset D V is a clique rdominating set of G if for every clique C is a vertex u D which rdominates C. A clique rpacking set is a subset P that there are no two distinct cliques C # ,C ## Prdominated by a common vertex of G. The clique rdomination problem is to find a clique rdominating set with minimum size and the clique rpacking problem is to find a clique rpacking set with maximum size. The formulated problems include many domination and cliquetransversalrelated problems as special cases. In this paper an efficient algorithm is proposed for solving these problems on dually chordal graphs which are a natural generalization of strongly chordal graphs. The efficient algorithm is mainly based on the tree structure and special vertex elimination orderings of dually chordal graphs. In some important particular cases where the algorithm works in linear time the obtained results generalize and improve known results on strongly chordal graphs.
Cop and Robber Games when the Robber can Hide and Ride
, 2011
"... In the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G =(V,E). The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber ..."
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Cited by 9 (4 self)
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In the classical cop and robber game, two players, the cop C and the robber R, move alternatively along edges of a finite graph G =(V,E). The cop captures the robber if both players are on the same vertex at the same moment of time. A graph G is called cop win if the cop always captures the robber after a finite number of steps. Nowakowski and Winkler [Discrete Math., 43 (1983), pp. 235–239] and Quilliot [Problèmes de jeux, de point fixe, de connectivité et de représentation sur des graphes, des ensembles ordonnés et des hypergraphes, Thèsededoctorat d’état, Université de Paris VI, Paris, 1983] characterized the copwin graphs as graphs admitting a dismantling scheme. In this paper, we characterize in a similar way the class CWFR(s, s ′)ofcopwin graphs in the game in which the robber and the cop move at different speeds s and s ′ , s ′ ≤ s. We also establish some connections between copwin graphs for this game with s ′ <sand Gromov’s hyperbolicity. In the particular case s =2ands ′ = 1, we prove that the class of copwin graphs is exactly the wellknown class of dually chordal graphs. We show that all classes CWFR(s, 1), s ≥ 3, coincide, and we provide a structural characterization of these graphs. We also investigate several dismantling schemes necessary or sufficient for the copwin graphs in the game in which the robber is visible only every k moves for a fixed integer k>1. In particular, we characterize the graphs which are copwin for any value of k.
Tree Spanners for Bipartite Graphs and Probe Interval Graphs
, 2003
"... A tree tspanner T in a graph G is a spanning tree of G such that the distance between every pair of vertices in T is at most t times their distance in G. The tree tspanner problem asks whether a graph admits a tree tspanner, given t. We first substantially strengthen the known results for bip ..."
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Cited by 8 (3 self)
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A tree tspanner T in a graph G is a spanning tree of G such that the distance between every pair of vertices in T is at most t times their distance in G. The tree tspanner problem asks whether a graph admits a tree tspanner, given t. We first substantially strengthen the known results for bipartite graphs. We prove that the tree tspanner problem is NPcomplete even for chordal bipartite graphs for t 5, and every bipartite ATEfree graph has a tree 3spanner, which can be found in linear time. The best known before results were NPcompleteness for general bipartite graphs, and that every convex graph has a tree 3spanner. We next focus on the tree tspanner problem for probe interval graphs and related graph classes. The graph classes were introduced to deal with the physical mapping of DNA. From a graph theoretical point of view, the classes are natural generalizations of interval graphs. We show that these classes are tree 7spanner admissible, and a tree 7spanner can be constructed in O(m log n) time.
Complexity Aspects of the Helly Property: Graphs and Hypergraphs
, 2009
"... In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property. A family of subsets has the Helly property when every subfamily thereof, formed by pairwise intersecting subsets, contains a common element. Many generalizations of this property exist which are r ..."
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Cited by 7 (2 self)
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In 1923, Eduard Helly published his celebrated theorem, which originated the well known Helly property. A family of subsets has the Helly property when every subfamily thereof, formed by pairwise intersecting subsets, contains a common element. Many generalizations of this property exist which are relevant to some fields of mathematics, and have several applications in computer science. In this work, we survey complexity aspects of the Helly property. The main focus is on characterizations of several classes of graphs and hypergraphs related to the Helly property. We describe algorithms for solving different problems arising from the basic Helly property. We also discuss the complexity of these problems, some of them leading to NPhardness results.
New Proofs for Strongly Chordal Graphs and Chordal Bipartite Graphs
, 2004
"... We give new proofs of wellknown characterizations of strongly chordal graphs and chordal bipartite graphs. The key ingredient is the dual hypertree structure for totally balanced hypergraphs. We also consider split graphs and threshold graphs. ..."
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Cited by 7 (0 self)
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We give new proofs of wellknown characterizations of strongly chordal graphs and chordal bipartite graphs. The key ingredient is the dual hypertree structure for totally balanced hypergraphs. We also consider split graphs and threshold graphs.