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Hyperbolicity and chordality of a graph
 2p + ⌊ε/2⌋, and d = 2p + ⌊ε/2⌋ + 1, which gives S1 = 2, S2 = 4p + 2 ⌊ε/2⌋, and so h(a, b, c, d) = ε − 2 − 2 ⌊ε/2⌋  ≤ 2. Let us now assume that S2 = max {S1, S2, S3}. Since S1 + S3 = 4p + ε, the
"... Let G be a connected graph with the usual shortestpath metric d. The graph G is δhyperbolic provided for any vertices x,y,u,v in it, the two larger of the three sums d(u,v) + d(x,y),d(u,x) + d(v,y) and d(u,y) + d(v,x) differ by at most 2δ. The graph G is kchordal provided it has no induced cycle ..."
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of length greater than k. Brinkmann, Koolen and Moulton find that every 3chordal graph is 1hyperbolic and that graph is not 1hyperbolic if and only if it contains one of two special graphs 2 as an isometric subgraph. For every k ≥ 4, we show that a kchordal graph must be ⌊ k 2 ⌋ k−2
Computing Treedepth Faster Than 2n
"... Abstract. A connected graph has treedepth at most k if it is a subgraph of the closure of a rooted tree whose height is at most k. We give an algorithm which for a given nvertex graph G, in time O(1.9602n) computes the treedepth of G. Our algorithm is based on combinatorial results revealing the ..."
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Abstract. A connected graph has treedepth at most k if it is a subgraph of the closure of a rooted tree whose height is at most k. We give an algorithm which for a given nvertex graph G, in time O(1.9602n) computes the treedepth of G. Our algorithm is based on combinatorial results revealing
Complexity of Generalized Colourings of Chordal Graphs
, 2008
"... The generalized graph colouring problem (GCOL) for a fixed integer k, and fixed classes of graphs P1,...,Pk (usually describing some common graph properties), is to decide, for a given graph G, whether the vertex set of G can be partitioned into sets V1,...,Vk such that, for each i, the induced subg ..."
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subgraph of G on Vi belongs to Pi. It can be seen that GCOL generalizes many natural colouring and partitioning problems on graphs. In this thesis, we focus on generalized colouring problems in chordal graphs. The structure of chordal graphs is known to allow solving many difficult combinatorial problems
Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits
 C. R. Acad. Sci. Paris Ser. I Math
, 2001
"... Abstract. We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolationrelated quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of t ..."
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Cited by 272 (9 self)
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of the crossing probabilities and Cardy’s formula. Then we prove existence, uniqueness, and conformal invariance of the continuum scaling limit. In this paper we study critical (p = pc = 1 2 1.
A Tourist Guide through Treewidth
 Acta Cybernetica
, 1993
"... A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find treedecompositions, algorithms that use treedecompositions to solve hard problems efficiently, graph minor theory, and some applications ..."
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Cited by 270 (22 self)
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A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find treedecompositions, algorithms that use treedecompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography.
Additive spanners for kchordal graphs
 In 5 th Italian Conference on Algorithms and Complexity (CIAC), volume 2653 of LNCS
, 2003
"... Abstract. In this paper we show that every chordal graph with n vertices and m edges admits an additive 4spanner with at most 2n−2 edges and an additive 3spanner with at most O(n · log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory, 13(198 ..."
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Cited by 10 (3 self)
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Abstract. In this paper we show that every chordal graph with n vertices and m edges admits an additive 4spanner with at most 2n−2 edges and an additive 3spanner with at most O(n · log n) edges. This significantly improves results of Peleg and Schäffer from [Graph Spanners, J. Graph Theory, 13
The cliqueseparator graph for chordal graphs
, 2007
"... We present a new representation of a chordal graph called the cliqueseparator graph, whose nodes are the maximal cliques and minimal vertex separators of the graph. We present structural properties of the cliqueseparator graph and additional properties when the chordal graph is an interval graph, ..."
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, proper interval graph, or split graph. We also characterize proper interval graphs and split graphs in terms of the cliqueseparator graph. We present an algorithm that constructs the cliqueseparator graph of a chordal graph in O(n³) time and of an interval graph in O(n²) time, where n is the number
Computing pathwidth faster than 2n?
"... Abstract. Computing the Pathwidth of a graph is the problem of finding a tree decomposition of minimum width, where the decomposition tree is a path. It can be easily computed in O∗(2n) time by using dynamic programming over all vertex subsets. For some time now there has been an open problem if th ..."
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Abstract. Computing the Pathwidth of a graph is the problem of finding a tree decomposition of minimum width, where the decomposition tree is a path. It can be easily computed in O∗(2n) time by using dynamic programming over all vertex subsets. For some time now there has been an open problem
Results 11  20
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2,707