Showing papers on "Split graph published in 2002"
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: The proposed MCES algorithm is based on a maximum clique formulation of the problem and is a significant improvement over other published algorithms and presents new approaches to both lower and upper bounding as well as vertex selection.
Abstract: A new graph similarity calculation procedure is introduced for comparing labeled graphs. Given a minimum similarity threshold, the procedure consists of an initial screening process to determine whether it is possible for the measure of similarity between the two graphs to exceed the minimum threshold, followed by a rigorous maximum common edge subgraph (MCES) detection algorithm to compute the exact degree and composition of similarity. The proposed MCES algorithm is based on a maximum clique formulation of the problem and is a significant improvement over other published algorithms. It presents new approaches to both lower and upper bounding as well as vertex selection.
327 citations
19 May 2002
TL;DR: It is shown that the recognition problem for string graphs is in NP, and therefore NP-complete, since Kratochvíl showed that the Recognition problem is NP-hard.
Abstract: A string graph is the intersection graph of a set of curves in the plane. Each curve is represented by a vertex, and an edge between two vertices means that the corresponding curves intersect. We show that string graphs can be recognized in NP. The recognition problem was not known to be decidable until very recently, when two independent papers established exponential upper bounds on the number of intersections needed to realize a string graph [18, 20]. These results implied that the recognition problem lies in NEXP. In the present paper we improve this by showing that the recognition problem for string graphs is in NP, and therefore NP-complete, since Kratochvil [12] showed that the recognition problem is NP-hard. The result has consequences for the computational complexity of problems in graph drawing, and topological inference.
149 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
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].
112 citations
TL;DR: This paper extends the study of quasi-randomness to sparse graphs, i.e., graphs on n vertices with o(n) edges, and it will be shown that many of the quasi- random properties for dense graphs have analogues for sparse graph, while others do not, at least not without additional hypotheses.
Abstract: Quasi-random graph properties form a large equivalence class of graph properties which are all shared by random graphs. In recent years, various aspects of these properties have been treated by a number of authors (e.g., see [5]-[14], [16], [23]-[27]). Almost all of these results deal with dense graphs, that is, graphs on n vertices having cn edges for some c > 0 as n → ∞. In this paper, we extend our study of quasi-randomness to sparse graphs, i.e., graphs on n vertices with o(n) edges. It will be shown that many of the quasi-random properties for dense graphs have analogues for sparse graphs, while others do not, at least not without additional hypotheses. In general, sparse graphs are more difficult to deal with than dense graphs, due for example to the possible absence of certain local structures, such as 4-cycles.
75 citations
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.
71 citations
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.
69 citations
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.
67 citations
21 Nov 2002
TL;DR: The paper presents several results that can be helpful for deciding whether the clique-width of graphs in a certain class is bounded or not, and applies these results to a number of particular graph classes.
Abstract: The paper presents several results that can be helpful for deciding whether the clique-width of graphs in a certain class is bounded or not, and applies these results to a number of particular graph classes.
66 citations
TL;DR: The chromatic number for graphs with forbidden induced subgraphs is studied for graph classes for which the question of 3-colourability can be decided in polynomial time and, if so, a proper 3-colouring can be determined also in poynomial time.
TL;DR: It is shown that the class of k-letter graphs is well-quasi-ordered by the induced sub graph relation, and that it has a finite set of minimal forbidden induced subgraphs.
TL;DR: It is shown that the Maximum Weight Stable Set (MWS) Problem can be solved in polynomial time when restricted to claw-free graphs and the structure of graphs being both claw- free and co-claw-free is very simple which implies bounded clique width for this graph class.
10 Aug 2002
TL;DR: A set of heuristics is developed which together allow the performance of semi-external DFS for directed graphs in practice, and is between 10 and 200 times faster than the best alternative, a factor that will further increase with future technological developments.
Abstract: Computing the strong components of a directed graph is an essential operation for a basic structural analysis of it. This problem can be solved by twice running a depth-first search (DFS). In an external setting, in which all data can no longer be stored in the main memory, the DFS problem is unsolved so far. Assuming that node-related data can be stored internally, semi-external computing does not make the problem substantially easier. Considering the definite need to analyze very large graphs, we have developed a set of heuristics which together allow the performance of semi-external DFS for directed graphs in practice. The heuristics have been applied to graphs with very different graph properties, including "web graphs" as described in the most recent literature and some large call graphs from ATT. Depending on the graph structure, the program is between 10 and 200 times faster than the best alternative, a factor that will further increase with future technological developments.
TL;DR: It is proved that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect, which leads to polynomial time algorithms for finding the parameters τC (G) and αC(G), for graphs belonging to this class.
Abstract: A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A clique-independent set is a subset of pairwise disjoint cliques of G. Denote by τ
C
(G) and α
C
(G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when τ
C
(H)=α
C
(H), for every induced subgraph H of G. In this paper, we prove that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect. The proof leads to polynomial time algorithms for finding the parameters τ
C
(G) and α
C
(G), for graphs belonging to this class. In addition, we prove that to decide whether or not a given subset of vertices of a graph is a clique-transversal is Co-NP-Complete. The complexity of this problem has been mentioned as unknown in the literature. Finally, we describe a family of highly clique-imperfect graphs, that is, a family of graphs G whose difference τ
C
(G)−α
C
(G) is arbitrarily large.
TL;DR: The dynamical behaviour of surface triangulations under the iterated application of the clique graph operator k, which transforms each graph G into the intersection graph kG of its (maximal) cliques is studied, showing that G is k-bounded for all d ?
TL;DR: A way to obtain all imperfect (P5, diamond)-free graphs by iterated point multiplication or substitution from a finite collection of small basic graphs is provided.
TL;DR: This work considers orientation problems on mixed graphs in which the goal is to obtain a directed graph satisfying certain connectivity requirements.
TL;DR: This paper considers graphs with maximum degree Δ = 3 and shows that the conjecture that every graph can be incidence-colored with Δ+2 colors holds for cubic Hamiltonian graphs and some other cubic graphs.
TL;DR: It is shown that the family of locally-quasiprimitive graphs is closed under the formation of a certain kind of quotient graph, called a normal quotient, induced by a normal subgroup, and the graph theoretic condition of local quAsiprimitivity is strictly weaker than the conditions of local primitivity and 2-arc transitivity.
TL;DR: It is shown that all clique trees of a chordal graph can be obtained from the reduced clique hypergraph; thus the reduced adjacency hypergraph can be thought of as a generalization of the notion of a clique tree.
TL;DR: Chordal triangle-free graphs are introduced as a natural superclass of chordal bipartite graphs and the structure of the maximal triangle- free members is described by describing their structure in terms of homomorphisms.
TL;DR: It is shown that with respect to toughness, a star graph is no better than an n-cube, whereas the alternating group graph and split-star are tougher than both of these graphs.
TL;DR: An important family of graphs is introduced which is closed under the clique operator and contains clique divergent graphs with strictly linear growth, i.e., o(knG) = o(G) + rn, where r is any fixed positive integer.
CEMA1
TL;DR: This work shows that the problem of finding in a graph a maximum weight induced path has polynomial time algorithms for k-bounded-hole families of graphs, for interval-filament graphs and for graphs decomposable by clique cut-sets or by splits into prime subgraphs for which such algorithms exist.
TL;DR: This paper shows that d k ⩽(2 n +3 k )/5 for all connected graphs of order n and minimum degree at least 2.5 improves on the Sanchis bound for dense graphs, namely thoseconnected graphs of size q and order n satisfying q >(6 n − k −5)/5.
TL;DR: It is shown that it is possible to recognize in polynomial time whether a given directed graph is an induced subgraph of some directed de Bruijn graph with given size of the labels.
TL;DR: The maximal exceptional graphs are determined by a computer search using the star complement technique, and it is shown how they can be found by theoretical considerations using a representation of E8 in R8.
01 Jan 2002
TL;DR: In this article, the authors proposed a new random graph model with given expected degree sequences, where each vertex is assigned to a weight (or the expected degree) and edges are assigned to each pairs independently with probability proportional to the product of their end-vertex-weights.
Abstract: To study the power law graph, we propose a new random graph model with given expected degree sequences. In this model, each vertex is assigned to a weight (or the expected degree) and edges are assigned to each pairs independently with probability proportional to the product of their end-vertex-weights. The classical random graph G(n,p) of Erdohs and Renyi can be viewed as a special case with the equal weights. We show that the distribution of the connected components depending primarily on the average degree d and the second-order average degree d˜. Here d˜ denotes the ratio of the sum of squares of the expected degree and the sum of the expected degrees of vertices. For example, we prove that the giant component exists if the expected average degree d is greater than 1, and there is no giant component if the expected second-order average degree d˜ is at most 1. Examples are given to illustrate that both bounds are best possible. We also obtain the tight upper bound on the size of the second largest connected components when d > 1. The existence of such a bound regardless the weight distribution is a quite amazing fact.
We further identify a family of the random graphs, whose average distance can be determined. These graphs are called “admissible”. The admissible graphs include G(n,p) and the power law graph with the power β > 3, and much more. For the admissible graph, we proved that almost surely the average distance is (1 + o(1)) lognlogdd . Similarly we examine the “strongly admissible” graph. We show that almost surely that the diameter of a strongly admissible graph is Θ( lognlogdd ). Again, this result applies to G(n,p) and the power law graph with the power β > 3.
The random power law graph is defined as a special random graph with power-law distribution weights. As we point out, the random power law graphs with the power β > 3 are both admissible and strongly admissible.
In the last chapter, we will focus on how the power law graphs are generated. We examine three important aspects of power law graphs, (1) the evolution of power law graphs, (2) the asymmetry of in-degrees and out-degrees, (3) the “scale invariance” of power law graphs. (Abstract shortened by UMI.)
TL;DR: An O(n2.376) algorithm for recognizing claw-free AT-free graphs and a linear-time algorithm for computing the set of all central vertices of a claw- free AT- free graph are presented.