Showing papers on "Chordal graph published in 2002"
TL;DR: An analytical expression for the cluster coefficient is derived, which shows that the graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality are distinctly different from standard random graphs, even for infinite dimensionality.
Abstract: We analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size of the largest cluster. We derive an analytical expression for the cluster coefficient, which shows that the graphs are distinctly different from standard random graphs, even for infinite dimensionality. Insights relevant for graph bipartitioning are included.
1,271 citations
TL;DR: A branch-and-bound algorithm for the maximum clique problem--which is computationally equivalent to the maximum independent (stable) set problem--is presented with the vertex order taken from a coloring of the vertices and with a new pruning strategy.
645 citations
TL;DR: It is proved that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree.
Abstract: We show that a number of graph-theoretic counting problems remain ${\cal NP}$-hard, indeed $\#{\cal P}$-complete, in very restricted classes of graphs. In particular, we prove that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. We obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2-CNF formulae. To achieve these results, a new interpolation-based reduction technique which preserves properties such as constant degree is introduced.
305 citations
TL;DR: It is shown that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs.
Abstract: We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs We prove that for all classes of graphs for which polynomial algorithms computing the treewidth and the minimum fill-in exist, we can list their potential maximal cliques in polynomial time Our approach unifies these algorithms Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs for which the treewidth and the minimum fill-in problems were open
214 citations
01 Jan 2002
TL;DR: This paper gives three increasingly general directed graph models and one general undirected graph model for generating power law graphs by adding at most one node and possibly one or more edges at a time and describes a method for scaling the time in the evolution model such that the power law of the degree sequences remains invariant.
Abstract: Many massive graphs (such as WWW graphs and Call graphs) share certain universal characteristics which can be described by the so-called the "power law" In this paper, we first briefly survey the history and previous work on power law graphs Then we give four evolution models for generating power law graphs by adding one node/edge at a time We show that for any given edge density and desired distributions for in-degrees and out-degrees (not necessarily the same, but adhered to certain general conditions), the resulting graph almost surely satisfy the power law and the in/out-degree conditions We show that our most general directed and undirected models include nearly all known models as special cases In addition, we consider another crucial aspect of massive graphs that is called "scale-free" in the sense that the frequency of sampling (wrt the growth rate) is independent of the parameter of the resulting power law graphs We show that our evolution models generate scale-free power law graphs
206 citations
TL;DR: It is found that graphs with low connectivity admit almost always a proper coloring, whereas graphs with high connectivity are uncolorable, and there exists a clustering phase c below c(q) in which ground states spontaneously divide into an exponential number of clusters.
Abstract: We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring, whereas graphs with high connectivity are uncolorable. Depending on q, we find the precise value of the critical average connectivity c q . Moreover, we show that below c q there exists a clustering phase c E [c d , c q ] in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms.
204 citations
TL;DR: The Moore bound is extended here to cover irregular graphs as well, yielding an affirmative answer to an old open problem.
Abstract: What is the largest number of edges in a graph of order n and girth g? For d-regular graphs, essentially the best known answer is provided by the Moore bound. This result is extended here to cover irregular graphs as well, yielding an affirmative answer to an old open problem ([4] p. 163, problem 10).
201 citations
TL;DR: An approach for the rule-based transformation of hierarchically structured hypergraphs, which extends the well-known double-pushout approach from flat to hierarchical graphs and makes rules more expressive by introducing variables which allow to copy and remove hierarchical subgraphs in a single rule application.
125 citations
01 Jan 2002
TL;DR: In this article, the authors consider the problem of minimizing the cost of an assignment in a n×n matrix of random costs and find the limit of the expected cost for the n cases.
Abstract: The random assignment problem is to minimize the cost of an assignment in a n×n matrix of random costs. In this paper we study this problem for some integer valued cost distributions. We consider both uniform distributions on 1, 2, . . . , m, for m = n or n, and random permutations of 1, 2, . . . , n for each row, or of 1, 2, . . . , n for the whole matrix. We find the limit of the expected cost for the n cases, and prove bounds for the n cases. This is done by simple coupling arguments together with Aldous recent results for the continuous case. We also present a simulation study of these cases.
115 citations
TL;DR: In this article, it was shown that the potential maximal cliques of a graph can be generated in polynomial time in the number of maximal separators of the graph, and that the treewidth and the minimum fill-in are polynomially tractable for all classes of graphs with a constant number of minimal separators.
01 Jan 2002
TL;DR: A vertex-magic total labeling of a graph with vertices and edges is defined as a one-to-one map taking the nodes and edges onto the integers $1, 2,..., v+e$ with the property that the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex.
Abstract: A vertex-magic total labeling of a graph with $v$ vertices and $e$ edges is defined as a one-to-one map taking the vertices and edges onto the integers $1, 2, . . . , v+e$ with the property that the sum of the label on a vertex and the labels on its incident edges is a constant independent of the choice of vertex.
Properties of these labelings are studied. It is shown how to construct labelings for several families of graphs, including cycles, paths, complete graphs of odd order and the complete bipartite graph $K_n,n$. It is also shown that labelings are impossible for some other classes of graphs.
TL;DR: The main result shows that for all 3 ≤ d ≤ n - 4 the random d-regular graph on n vertices almost surely has no nontrivial automorphisms.
Abstract: This paper studies the symmetry of random regular graphs and random graphs. Our main result shows that for all 3 ≤ d ≤ n - 4 the random d-regular graph on n vertices almost surely has no nontrivial automorphisms. This answers an open question of N. Wormald [13].
TL;DR: A new polynomially solvable case for the problem in bipartite graphs which deals with a generalization of bi-complement reducible graphs is described.
TL;DR: It is proved that regular graphs with order more than 5 have at least one 3-RC, and it is shown that vertex-and edge-transitive graphs other than cycles are super-λ(3).
TL;DR: The asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges is derived and it is shown that almost all such graphs with $n$ vertices contain many copies of any fixed planar graph, and this implies that almostAll such graphs have large automorphism groups.
Abstract: We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with $n$ vertices contain many copies of any fixed planar graph, and this implies that almost all such graphs have large automorphism groups.
TL;DR: Some large classes of graphs, such as all connected edge- transitive graphs except a star, and all connected vertex-transitive graphs with odd order or without triangles, have equality of the restricted edge-connectivity and the minimun edge-degree.
Book•
01 Nov 2002
TL;DR: In this paper, the authors introduce the concepts of Graphs and their uses, such as distance and connectivity, graph coloring, and graph algorithms, and planar graphs.
Abstract: 1. Introductory Concepts. 2. Introduction to Graphs and their Uses. 3. Trees and Bipartite Graphs. 4. Distance and Connectivity. 5. Eularian and Hamiltonian Graphs. 6. Graph Coloring. 7. Matrices. 8. Graph Algorithms. 9. Planar Graphs. 10. Digraphs and Networks. 11. Special Topics. Answers/Solutions to Selected Exercises. Index.
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TL;DR: An NP characterization is given for planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires).
Abstract: We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give an NP characterization for such graphs, derive some consequences regarding sparsity and coloring, and survey some algorithmic results.
16 Nov 2002
TL;DR: An algorithm that broadcasts in logarithmic time on all graphs from the work of Bar-Yehuda et al. is constructed, giving the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting.
Abstract: In a seminal paper, Bar-Yehuda et al. (1992) considered broadcasting in radio networks whose nodes know only their own label and labels of their neighbors. They claimed a linear lower bound on the time of deterministic broadcasting in such radio networks, by constructing a class of graphs of diameter 3, with the property that every broadcasting algorithm requires linear time on one of these graphs. Due to a subtle error in the argument, this result is incorrect. We construct an algorithm that broadcasts in logarithmic time on all graphs from the work of Bar-Yehuda et al. Moreover, we show how to broadcast in sublinear time on all n-node graphs of diameter o(log log n). On the other hand, we construct a class of graphs of diameter 4, such that every broadcasting algorithm requires time /spl Omega/(4/spl radic/n) on one of these graphs. In view of the randomized algorithm, running in expected time O(D log n + log/sup 2/ n) on all n-node graphs of diameter D, our lower bound gives the first correct proof of an exponential gap between determinism and randomization in the time of radio broadcasting.
TL;DR: In this article, the bicolorability of a clique hypergraph C(G) is studied and the question of whether the vertices of G can be colored with two colors so that no maximal clique is monochromatic is solved in polynomial time.
19 May 2002
TL;DR: This work considers the problem of 3-coloring sparse random graphs and analyzes a "smoothed" version of the Brelaz heuristic to prove that almost all graphs with average degree d, i.e. G(n,p=d/n), are 3-colorable for d/n.
Abstract: The technique of using differential equations to approximate the mean path of Markov chains has proved very useful in the average-case analysis of algorithms. Here, we significantly expand the range of this technique, by showing that it can be used to handle algorithms that favor high-degree vertices. In particular, we consider the problem of 3-coloring sparse random graphs and analyze a "smoothed" version of the Brelaz heuristic. This allows us to prove that i) almost all graphs with average degree d, i.e. G(n,p=d/n), are 3-colorable for d≤ 4.03, and that ii) almost all 4-regular graphs are 3-colorable. This improves over the previous lower bound of 3.847 for the G(n,p) 3-colorability threshold and gives the first non trivial result on the 3-colorability of random regular graphs.
26 Aug 2002
TL;DR: A polynomial time algorithm for c-planarity testing of "almost" c-connected clustered graphs, i.e., graphs for which all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T.
Abstract: A clustered graph C = (G, T) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the vertices of G = (V, E). Each vertex µ in T corresponds to a subset of the vertices of the graph called "cluster". c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automaticgraph drawing. The complexity status of c-planarity testing is unknown. It has been shown in [FCE95, Dah98] that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected.In this paper, we provide a polynomial time algorithm for c-planarity testing of "almost" c-connected clustered graphs, i.e., graphs for which all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each nonconnected cluster its super-cluster and all its siblings in T are connected. The algorithm is based on the concepts for the subgraph induced planar connectivity augmentation problem presented in [GJL+02]. We regard it as a first step towards general c-planarity testing.
01 Jan 2002
TL;DR: In this paper, a polynomial time algorithm for c-planarity testing of almost-c-connected clustered graphs was proposed, i.e., graphs for which all nodes corresponding to the non-connected clusters lie on the same path in T starting at the root of T, or graphs in which for each nonconnected cluster its super-cluster and all its siblings in T are connected.
Abstract: A clustered graph C =( G, T ) consists of an undirected graph G and a rooted tree T in which the leaves of T correspond to the ver- tices of G =( V, E). Each vertex µ in T corresponds to a subset of the vertices of the graph called "cluster". c-planarity is a natural extension of graph planarity for clustered graphs, and plays an important role in automaticgraph drawing. The c omplexity status of c-planarity testing is unknown. It has been shown in (FCE95,Dah98) that c-planarity can be tested in linear time for c-connected graphs, i.e., graphs in which the cluster induced subgraphs are connected. In this paper, we provide a polynomial time algorithm for c-planarity testing of "almost" c-connected clustered graphs, i.e., graphs for which all nodes corresponding to the non-c-connected clusters lie on the same path in T starting at the root of T , or graphs in which for each non- connected cluster its super-cluster and all its siblings in T are connected. The algorithm is based on the concepts for the subgraph induced planar connectivity augmentation problem presented in (GJL + 02). We regard it as a first step towards general c-planarity testing.
22 May 2002
TL;DR: This paper identifies the special characteristics and problematics of such graphs and offers several algorithms for tackling them, viewed as carefully constructed extensions of force-directed methods, and their output quality and performance are similar.
Abstract: The vertices of most graphs that appear in real applications are non-uniform. They can be circles, ellipses, rectangles, or other geometric elements of varying shapes and sizes. Unfortunately, current force directed methods for laying out graphs are suitable mostly for graphs whose vertices are zero-sized and dimensionless points. It turns out that naively extending these methods to handle non-uniform vertices results in serious deficiencies in terms of output quality and performance. In this paper we try to remedy this situation by identifying the special characteristics and problematics of such graphs and offering several algorithms for tackling them. The algorithms can be viewed as carefully constructed extensions of force-directed methods, and their output quality and performance are similar.
TL;DR: It is shown that planar graphs without 3-cycles are 3-degenerate, and more surprisingly, that the same holds for planar graph without 6-cycles.
Abstract: It is easy to see that planar graphs without 3-cycles are 3-degenerate. Recently, it was proved that planar graphs without 5-cycles are also 3-degenerate. In this paper it is shown, more surprisingly, that the same holds for planar graphs without 6-cycles.
TL;DR: It is shown that the perfect (efficient) edge domination problem is NP-complete on bipartite (planar bipartites) graphs and linear-time algorithms to solve the weighted perfect ( efficient) edge dominating problem on generalized series-parallel graphs and chordal graphs are presented.
TL;DR: It is proved that, with probability tending to 1 as n → ∞, Gr is r-connected and Hamiltonian.
Abstract: Let Gr denote a graph chosen uniformly at random from the set of r-regular graphs with vertex set l1,2, …, nr, where 3 l r l c0n for some small constant c0. We prove that, with probability tending to 1 as n → ∞, Gr is r-connected and Hamiltonian.
TL;DR: This paper starts the study of computational complexity of locally injective homomorphisms called partial covers of graphs by showing a correspondence to generalized (2,1)-colorings of graphs, stemming from a practical problem of assigning frequencies to transmitters without interference.
Abstract: Given graphs G and H, a mapping f : V (G) → V (H) is a homomorphism if (f(u), f(v)) is an edge of H for every edge (u, v) of G. In this paper, we initiate the study of computational complexity of locally injective homomorphisms called partial covers of graphs. We motivate the study of partial covers by showing a correspondence to generalized (2,1)-colorings of graphs, the notion stemming from a practical problem of assigning frequencies to transmitters without interference. We compare the problems of deciding existence of partial covers and of full covers (locally bijective homomorphisms), which were previously studied.
TL;DR: This work uses multi-terminal networks to unify known constructions of snarks (nontrivial cubic graphs without edge- 3-colorings) and provide new ones in the same process, implying new complexity results about nowhere-zero flows in graphs.
Abstract: Using multi-terminal networks we build methods on constructing graphs without nowhere-zero group- and integer-valued flows. In this way we unify known constructions of snarks (nontrivial cubic graphs without edge- 3-colorings, or equivalently, without nowhere-zero 4-flows) and provide new ones in the same process. Our methods also imply new complexity results about nowhere-zero flows in graphs and state equivalences of Tutte?s 3- and 5-flow conjectures with formally weaker statements.