Showing papers on "Split graph published in 1993"
TL;DR: It is shown that the edge set of a cubic graph can always be partitioned into 10 subsets, each of which induces a matching in the graph.
Abstract: In this paper, we show that the edge set of a cubic graph can always be partitioned into 10 subsets, each of which induces a matching in the graph. This result is a special case of a general conjecture made by Erdos and NeSetiil: For each d 2 3, the edge set of a graph of maximum degree d can always be partitioned into [5d2/4] subsets each of which induces a matching. 0 1993 John Wiley & Sons, Inc.
144 citations
TL;DR: An O( nm )-time algorithm to decompose a graph with n vertices and m edges by means of clique separators is described, modified, so that no new maximal prime subgraphs are generated, i.e. so that a graph is decomposed exactly into its maximal primeSubgraphs which is the unique minimal derived system of prime sub graphs.
130 citations
TL;DR: In this article, two variants of a tabu search heuristic, a deterministic one and a probabilistic one, for the maximum clique problem are described and compared with those of other approximate methods.
Abstract: We describe two variants of a tabu search heuristic, a deterministic one and a probabilistic one, for the maximum clique problem. This heuristic may be viewed as a natural alternative implementation of tabu search for this problem when compared to existing ones. We also present a new random graph generator, the\(\hat p\)-generator, which produces graphs with larger clique sizes than comparable ones obtained by classical random graph generating techniques. Computational results on a large set of test problems randomly generated with this new generator are reported and compared with those of other approximate methods.
123 citations
TL;DR: It is shown that the pathwidth problem for chordal graphs is NP-hard, and polynomial algorithms for subclasses are given – a generalization of split graphs and a class of graphs where the intersection behavior of maximal cliques is strongly restricted.
98 citations
TL;DR: The structure of graphs containing neither the 4-cycle nor its complement as an induced subgraph is characterized, which includes split graphs, which are graphs whose vertex set is the union of a clique and an independent set.
70 citations
01 Jan 1993
TL;DR: An improved random graph generation system, DELTA, is presented, which increases the range of maximal independent sets that can be hidden from attack using quite simple techniques that are successful signiicantly often for graphs of practical size.
Abstract: In this paper, we look at the problem of how one might try to hide a large independent set in a graph in which all other independent sets are signiicantly smaller. We observe that the most common methods of generating graphs with known maximal independent sets are subject to attack using quite simple techniques that are successful signiicantly often for graphs of practical size. We present an improved random graph generation system, DELTA, which increases the range of maximal independent sets that can be hidden from these techniques.
57 citations
TL;DR: It is shown that these four problems are NP-complete in split graphs with degree constraints and linear time algorithms for them are given in a strongly chordal graph when a strong elimination order is given.
Abstract: In a graph $G = ( V,E ),E [ v ]$ denotes the set of edges in the subgraph induced by $N [ v ] \equiv \{ v \} \cup \{ u \in V:uv \in E \}$. The neighborhood-covering problem is to find the minimum cardinality of a set C of vertices such that $E = \cup \{ E [ v ]:v \in C \}$. The neighborhood-independence problem is to find the maximum cardinality of a set of edges in which there are no two distinct edges belonging to the same $E [ v ]$ for any $v \in V$. Two other related problems are the clique-transversal problem and the clique-independence problem. It is shown that these four problems are NP-complete in split graphs with degree constraints and linear time algorithms for them are given in a strongly chordal graph when a strong elimination order is given.
49 citations
16 Jun 1993
TL;DR: Recently in several papers independently graphs with maximum neighbourhood orderings were characterized and turned out to be algorithmically useful.
Abstract: Recently in several papers ([10],[22],[42]) independently graphs with maximum neighbourhood orderings were characterized and turned out to be algorithmically useful.
43 citations
01 Mar 1993
TL;DR: A precise call graph enables data flow optimizations and for understanding programs and the importance of precisely constructing an analogue of call graph in the context of higher order languages such as Scheme and ML has been eloquently elaborated by Shivers.
Abstract: A call m.ultigraph’of a program is a directed Multigraph encoding the possible calling relations between procedures. These graphs are used in interprocedurd program optimization [2, 3, 9, 15] and for reverse engineering of softw~are systems [7, 8]. For programs that do not contain proCedul”e valued variables (referred to hencefollh as procedure variables) this graph can be constructed by a single pass over the program collecting the procedures called at each call site. When procedure v,ari.ables and indirect calls using values of such variables are allowed constructing such a graph is not so simple. In the worst case, the value of a procedure v,ariable at a call site may be a reference to any procedure in the program. For interprocedural optimizations and for understanding programs one would like to have more precise solutions. The importance of precisely constructing an analogue of call graph (referred to as the Ot}l order control flow analysis or OCFA) in the context of higher order languages such as Scheme and ML has been eloquently elaborated by Shivers [18]. A precise call graph enables data flow optimizations * In ths paper call multigmph is also refereed to as the call graph. Ths work was supported by the grant LEQSF (1991-92) ENH-98 from the Louisiana Bowl of Regents.
43 citations
TL;DR: It is proved that the stability number of bull-free chair-free graphs can be found in polynomial time.
43 citations
TL;DR: Efficient algorithms for computing R(G) of various restricted classes of networks that admit polynomial time algorithms include trees, series-parallel graphs, partial k-trees, directed path graphs and permutation graphs are presented.
15 Dec 1993
TL;DR: In this paper, the treewidth of a circle graph can be computed in polynomial time, where n is the number of vertices of the graph and isomorphic to the intersection graph of a finite collection of chords.
Abstract: In this paper we show that the treewidth of a circle graph can be computed in polynomial time. A circle graph is a graph that is isomorphic to the intersection graph of a finite collection of chords of a circle. The TREEWIDTH problem can be viewed upon as the problem of finding a chordal embedding of the graph that minimizes the clique number. Our algorithm to determine the treewidth of a circle graph can be implemented to run in O(n3) time, where n is the number of vertices of the graph.
TL;DR: A strengthening of a lemma due to A. Ben Rebea is presented and it is shown that every vertex in a claw-free graph with stability number at least three is such that its neighbourhood can be covered by two complete graphs or contains an induced C5.
TL;DR: It is shown that partition graphs are preserved by removal of any vertex whose closed neighborhood properly contains the closed neighborhood of some other vertex.
TL;DR: The maximum number of maximal independent sets among all bipartite graphs of order n and the extremal graphs as well as the corresponding results for connected bipartites graphs are determined.
Abstract: A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G In this paper, we determine the maximum number of maximal independent sets among all bipartite graphs of order n and the extremal graphs as well as the corresponding results for connected bipartite graphs © 1993 John Wiley & Sons, Inc
TL;DR: The results are interesting because most of other variants of the domination problem are NP-complete on both chordal graphs and split graphs, except for the independent domination problem.
TL;DR: The purpose in the present paper is to investigate different topological properties of the recently introduced star graphs which are being viewed as attractive alternatives to n -cubes or hypercubes.
Abstract: Our purpose in the present paper is to investigate different topological properties of the recently introduced star graphs which are being viewed as attractive alternatives to n -cubes or hypercubes. These properties are interesting by themselves from a graph theory point of view as well as they have direct in generating vertex disjoint paths and minimal paths in a star graph. They can also be readily utilized to design routing algorithms and to compute contention and traffic congestion in networks that use star graphs as the underlying topology.
Posted Content•
TL;DR: In this paper, the authors determined all graphs with the spectrum of a distance-regular graph with at most 30 vertices, except possibly for the Taylor graph on 28 vertices.
Abstract: We determine all graphs with the spectrum of a distance-regular graph with at most 30 vertices (except possibly for the Taylor graph on 28 vertices)
01 Jan 1993
TL;DR: It is proved that it remains NP-complete when the input is restricted to the classes of split graphs or graphs with maximum degree 3 k +2, and it is shown that the SPANNING 2-TREE problem remains NP -complete when restricted to planar graphs withmaximum degree at most 6.
TL;DR: Using a result of Ne?et?il and Rodl that Hasse diagram orientation is NP-complete, this gives a new proof for NP-hardness of the string graph recognition problem.
TL;DR: Two questions which arose in the study of partition graphs are answered by recently discovered examples and an enumeration of the partition graphs on ten or fewer vertices is provided.
TL;DR: An intrinsic characterization of extremal interval graphs is given and recurrence relations for the numbers of such graphs are derived.
Abstract: An interval graph is said to be extremal if it achieves, among all interval graphs having the same number of vertices and the same clique number, the maximum possible number of edges. We give an intrinsic characterization of extremal interval graphs and derive recurrence relations for the numbers of such graphs. © 1993 John Wiley & Sons, Inc.
03 May 1993
TL;DR: A parallel algorithm based on the maximum neural network which resembles the winner-takes-all circuit is introduced which solves large-scale problems in reasonable computation time that the best known algorithms cannot solve.
Abstract: The maximum clique problem is to find the maximum complete subgraph of a given graph G. A computation model for large-scale maximum clique problems is proposed and was tested. A parallel algorithm based on the maximum neural network which resembles the winner-takes-all circuit is introduced which solves large-scale problems in reasonable computation time that the best known algorithms cannot solve. The maximum clique problem is first formulated as an unconstrained quadratic zero-one programming problem and is solved by minimizing the weight summation over the same partition in a newly constructed graph. >
TL;DR: A new class of graphs for which the stability number can be obtained in polynomial time is described, based on an iterative procedure that builds from a graph G a new graph Gl that has fewer nodes and has the property that α(Gl) = α(G) − 1.
Abstract: We describe a new class of graphs for which the stability number can be obtained in polynomial time. The algorithm is based on an iterative procedure that, at each step, builds from a graph G a new graph Gl that has fewer nodes and has the property that α(Gl) = α(G) − 1.
This new class of graphs is different from the known classes for which the stability number can be computed in polynomial time. © 1993 John Wiley & Sons, Inc.
Book•
01 Mar 1993
TL;DR: Tutte et al. as mentioned in this paper gave a survey of (m,k)-colorings, M.T. Farrell gave a note on well-covered graphs, V. Zhang matching extensions and productos of graphs, J. Liu and Q. Yu prospects for graph theory algorithms, R. Steinberg ranking planar embeddings using PQ-trees, A. Widulski the random f-graph process.
Abstract: Whither graph theory?, W.T. Tutte the future of graph theory, B. Bollobas new directions in graph theory (with an emphasis on the role of applications), F.S. Roberts a survey of (m,k)-colorings, M. Frick numerical decks of trees, F. Gavril et al the complexity of colouring by infinite vertex transitive graphs, B. Bauslaugh rainbow subgraphs in edge-colourings of complete graphs, P. Erdos and Z. Tuza graphs with special distance properties, M. Lewinter probability models for random multigraphs with applications in cluster analysis, E.A.J. Godehardt solved and unsolved problems in chemical graph theory, A.T. Balaban detour distance in graphs, G. Chartrand et al integer-distance graphs, R.P. Grimaldi toughness and the cycle structure of graphs, D. Bauer and E, Schmeichel the Birkhoff-Lewis equations for graph-colourings, W.T. Tutte the complexity of knots, D.J.A. Welsh the impact of F-polynomials in graph theory, E.J. Farrell a note on well-covered graphs, V. Chvatal and P.J. Slater cycle covers and cycle decomposition of graphs, C.-Q. Zhang matching extensions and productos of graphs, J. Liu and Q. Yu prospects for graph theory algorithms, R.C. Read the state of the three colour problem, R. Steinberg ranking planar embeddings using PQ-trees, A. Karabeg some problems and results in cochromatic theory, P. Erdos and J. Gimbel from random graphs to graph theory, A. Rucinski matching and vertex packing - how "hard" are they?, M.D. Plummer the competition number and its variants, S.-R. Kim which double starlike trees span ladders?, M. Lewinter and W.F. Widulski the random f-graph process, K.T. Balinska and L.V. Quintas quo vadis, random graph theory?, E.M. Palmer exploratory statistical analysis of networks, O. Frank and K. Norwicki the Hamiltonian decomposition of certain circulant graphs, J. Liu discovery-method teaching in graph theory, P.Z. Chinn.
TL;DR: In this paper, an O(n log n + k) time algorithm is presented to generate all maximal independent sets in a permutation graph, where n is the number of vertices in the graph and k is the Number of Vertices generated.
Abstract: In this paper, an O(n log n + k) time algorithm is presented to generate all maximal independent sets in a permutation graph, where n is the number of vertices in the graph and k is the number of vertices generated. The space requirement is O(n log n).
Dissertation•
02 Jan 1993TL;DR: This thesis presents a practical, efficient linear algorithm to find a 4-elimination sequence for a graph of tree-width at most four, which depends upon the graphs having certain reductions as well.
Abstract: A graph G has tree-width at most k if the vertices of G can be decomposed into a tree-like structure of sets of vertices, each set having cardinality at most $k+1.$ An alternate definition of tree-width is in terms of k-elimination sequences, an order to eliminate the vertices of the graph such that each vertex, at the time it is eliminated from the graph, has degree at most k. Amborg and Proskurowski have shown that if a graph has tree-width at most a fixed k, then many NP-hard problems can be solved in linear time, provided this k-elimination sequence is part of the input. Some examples are maximum independent set, minimum dominating set, chromatic number, Hamilton circuit, and network reliability. These algorithms are very efficient for small k, such as 2, 3, or 4, but may be impractical for large k, as they depend exponentially on k.
This thesis presents a practical, efficient linear algorithm to find a 4-elimination sequence for a graph of tree-width at most four. Graphs of tree-width at most k are a natural generalization of series-parallel graphs. Just as linear algorithms on series-parallel graphs depend upon their structure, that they have an instance of a series-reduction or a parallel-reduction, the algorithm presented in this thesis depends upon the graphs having certain reductions as well. A reduction process is developed, and reductions are shown that can be applied to a graph of tree-width at most four without increasing its tree-width. Further, each graph of tree-width at most four is shown to contain one of these reductions. The reductions are then used in a linear time algorithm which generates a 4-elimination sequence, if one exists.
TL;DR: The Perfect Graph Conjecture will be proved for 3-line graphs and 3-total graphs and perfect 3- line graphs are not contained in any of the known classes of perfect graphs.
Abstract: The concept of the line graph can be generalized as follows. The k-line graph Lk(G) of a graph G is defined as a graph whose vertices are the complete subgraphs on k vertices in G. Two distinct such complete subgraphs are adjacent in Lk(G) if and only if they have in G k − 1 vertices in common. The concept of the total graph can be generalized similarly. Then the Perfect Graph Conjecture will be proved for 3-line graphs and 3-total graphs. Moreover, perfect 3-line graphs are not contained in any of the known classes of perfect graphs. © 1993 John Wiley & Sons, Inc.
TL;DR: In this article, the authors explore the relationship between vertex neighborhoods N"u in a graph, focusing on neighborhood equality and inclusion (suitably modified for adjacent vertices) and characterize M(G and D(G) for threshold graphs and for trees.