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"... A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (M ..."
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Cited by 88 (6 self)
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A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the socalled Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several different ways. This allows us obtain several linear time multisweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2sweep MNS certifying recognition algorithm. Submitted:
Distributional limits for critical random graphs
 In preparation
, 2009
"... We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metri ..."
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Cited by 31 (8 self)
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We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed λ ∈ R. Then, as a metric space with the graph distance rescaled by n −1/3, the sequence of connected components G(n, p) converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean. Keywords: Random graphs, GromovHausdorff distance, scaling limits, continuum random tree, diameter. 2000 Mathematics subject classification: 05C80, 60C05.
Distributed computing of efficient routing schemes in generalized chordal graphs
, 2009
"... Efficient algorithms for computing routing tables should take advantage of the particular properties arising in large scale networks. There are in fact at least two properties that any routing scheme must consider: low (logarithmic) diameter and high clustering coefficient. High clustering coefficie ..."
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Cited by 13 (4 self)
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Efficient algorithms for computing routing tables should take advantage of the particular properties arising in large scale networks. There are in fact at least two properties that any routing scheme must consider: low (logarithmic) diameter and high clustering coefficient. High clustering coefficient implies the existence of few large induced cycles. Therefore, we propose a routing scheme that computes short routes in the class of kchordal graphs, i.e., graphs with no chordless cycles of length more than k. We study the tradeoff between the length of routes and the time complexity for computing them. In the class of kchordal graphs, our routing scheme achieves an additive stretch of at most k − 1, i.e., for all pairs of nodes, the length of the route never exceeds their distance plus k − 1. In order to compute the routing tables of any nnode graph with diameter D we propose a distributed algorithm which uses messages of size O(log n) and takes O(D) time. We then propose a slightly modified version of the algorithm for computing routing tables in time O(min{∆D, n}), where ∆ is the the maximum degree of the graph. Using these tables, our routing scheme achieves a better additive stretch of 1 in chordal graphs (notice that chordal graphs are 3chordal graphs). The routing scheme uses addresses of size log n bits and local memory of size 2(d − 1)log n bits per node with degree d.
A simple polynomial algorithm for the longest path problem on cocomparability graphs
, 2012
"... Given a graph G, the longest path problem asks to compute a simple path of G with the largest number of vertices. This problem is the most natural optimization version of the wellknown and wellstudied Hamiltonian path problem, and thus it is NPhard on general graphs. However, in contrast to the ..."
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Cited by 4 (2 self)
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Given a graph G, the longest path problem asks to compute a simple path of G with the largest number of vertices. This problem is the most natural optimization version of the wellknown and wellstudied Hamiltonian path problem, and thus it is NPhard on general graphs. However, in contrast to the Hamiltonian path problem, there are only a few restricted graph families, such as trees, and some small graph classes where polynomial algorithms for the longest path problem have been found. Recently it has been shown that this problem can be solved in polynomial time on interval graphs by applying dynamic programming to a characterizing ordering of the vertices of the given graph [K. Ioannidou, G. B. Mertzios, and S. D. Nikolopoulos, Algorithmica, 61 (2011), pp. 320–341], thus answering an open question. In the present paper, we provide the first polynomial algorithm for the longest path problem on a much greater class, namely on cocomparability graphs. Our algorithm uses a similar, but essentially simpler, dynamic programming approach, which is applied to a lexicographic depth first search (LDFS) characterizing ordering of the vertices of a cocomparability graph. Therefore, our results provide evidence that this general dynamic programming approach can be used in a more general setting, leading to efficient algorithms for the longest path problem on greater classes of graphs. LDFS has recently been introduced in
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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Cited by 2 (0 self)
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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 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, such as the graph colouring, maximum clique and others, in polynomial, and in many cases in linear time. Our study of generalized colouring problems focuses on those problems in which the sets Pi are characterized by a single forbidden induced subgraph. We show, that for k = 2, all such problems where the forbidden graphs have at most three vertices are polynomial time solvable in chordal graphs, whereas, it is known that almost all of them are NPcomplete in general. On the other hand, we show infinite families of such problems which are NPcomplete in chordal graphs. By combining a polynomial algorithm and an
A foursweep LBFS recognition algorithm for interval graphs
 Submitted. URL: http://math.sjtu.edu.cn/faculty/ykwu/ 4sweepLW.pdf
"... In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic BreadthFirst Search (LBFS) and prove its correctness. They believe that their corresponding 5sweep LBFS interval graph recognition algorithm is also correct. Thanks to ..."
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In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic BreadthFirst Search (LBFS) and prove its correctness. They believe that their corresponding 5sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.
To cite this version:
, 2010
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Article Graph Extremities Defined by Search Algorithms
, 2010
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.