Showing papers on "Split graph published in 1996"
TL;DR: In this paper, recognizability is understood in an algebraic sense, relative to a finite set of graph operations and basic graphs that generate all graphs of C, the modular decomposition of which uses prime graphs of bounded size.
177 citations
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19 Feb 1996TL;DR: In this paper, the authors studied the chromatic number of planar graphs and gave an upper bound for the total chromatic numbers of graphs, including complete r-partite graphs, graphs of low degree and graphs of high degree.
Abstract: Basic terminology and introduction.- Some basic results.- Complete r-partite graphs.- Graphs of low degree.- Graphs of high degree.- Classification of type 1 and type 2 graphs.- Total chromatic number of planar graphs.- Some upper bounds for the total chromatic number of graphs.- Concluding remarks.
146 citations
TL;DR: It is shown that the question “Is a graph 3-colorable?” remains NP-complete when restricted to the class of triangle-free graphs with maximum degree 4.
134 citations
TL;DR: Some new equivalent definitions of a quasi-threshold graph are given and linear time recognition algorithms follow that give linear time algorithms for the edge domination problem and the bandwidth problem in this class of graphs.
111 citations
TL;DR: It is found out that there are no genuine “global” pseudo-distance-regular graphs: when pseudo- Distance-Regularity is shared by all the vertices, the graph turns out to be distance-regular.
86 citations
TL;DR: In this article, it was shown that the vertex set of a finite undirected graph can be partitioned into one or two independent sets and two cliques in polynomial time.
82 citations
TL;DR: Here it is investigated the complexity status of precolouring extendibility on some classes of perfect graphs, giving good characterizations that lead to algorithms with linear or polynomial running time.
Abstract: We continue the study of the following general problem on the vertex colourings of graphs. Suppose that some vertices of a graph G are assigned to some colours. Can this ‘precolouring’ be extended to a proper colouring of G with at most k colours (for some given k)? Here we investigate the complexity status of precolouring extendibility on some classes of perfect graphs, giving good characterizations (necessary and sufficient conditions) that lead to algorithms with linear or polynomial running time. It is also shown how a larger subclass of perfect graphs can be derived from graphs containing no induced path on four vertices.
80 citations
TL;DR: It is shown that every 3 2 - tough split graph is hamiltonian and that there is a sequence of non-hamiltonian split graphs with toughness converging to 3 2 .
64 citations
TL;DR: In this paper, a graph G is called well covered if every maximal independent set of G has the same number of vertices, where vertices correspond to two maximal independent sets of G. In this paper, we extend the definition of well covered simplicial, chordal and circular arc graphs.
Abstract: A graph G is called well covered if every two maximal independent sets of G have the same number of vertices. In this paper, we characterize well covered simplicial, chordal and circular arc graphs. © 1996 John Wiley & Sons, Inc.
63 citations
01 Jul 1996
TL;DR: A linear time algorithm is presented that either finds an embedding of G in S or identifies a subgraph of G that is homomorphic to a minimal forbidden subgraph for embeddability in S that yields a constructive proof of the result of Robertson and Seymour that for each closed surface there are only finitely many minimal forbiddenSubgraphs.
Abstract: For an arbitrary fixed surface S, a linear time algorithm is presented that for a given graph G either finds an embedding of G in S or identifies a subgraph of G that is homomorphic to a minimal forbidden subgraph for embeddability in S. A side result of the proof of the algorithm is that minimal forbidden subgraphs for embeddability in S cannot be arbitrarily large. This yields a constructive proof of the result of Robertson and Seymour that for each closed surface there are only finitely many minimal forbidden subgraphs. The results and methods of this paper can be used to solve more general embedding extension problems.
56 citations
TL;DR: The classes of graphs (finite and infinite) of bounded tree-partition-width in terms of excluded topological minors are characterized.
TL;DR: It is proved that it is always possible to substitute some of the vertices of a non-perfect graph by cliques so that the resulting graph is not kernel solvable.
TL;DR: Algorithms for finding the clique transversal number and theClique independence number for a comparability graph of n nodes are presented, where M ( n ) is the complexity of multiplying two n × n matrices.
TL;DR: The first efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordalist graph, and finding a breadth-first and depth-first search tree are given.
Abstract: We give the first efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordal graph, finding a breadth-first search tree and a depth-first search tree of a chordal graph, recognizing interval graphs, and testing interval graphs for isomorphism. The key to our results is an efficient parallel algorithm for finding a perfect elimination ordering.
TL;DR: The d-convex sets in a metric space are those subsets which include the metric interval between any two of its elements, which entails a Helly theorem for quasi-median graphs, pseudo-modular graphs, and bridged graphs.
TL;DR: The Four Color Theorem can be equivalently stated in terms of anti-Gallai graphs; the problems of determining the clique number, and the chromatic number of a Gallai graph are NP-complete.
TL;DR: Complex results for the generalized clique transversal problem on subclasses of chordal graphs, e.g., strongly chordal graph graphs, k-trees, split graphs, and undirected path graphs are given.
TL;DR: This paper studies the structure of trapezoid graphs and shows that an operation called vertex splitting allows a trapezoids graph to be transformed into a permutation graph with special properties.
TL;DR: This work disproves a conjecture of Mulder on star contraction of median graphs and considers the analogous class of graphs associated with acyclic cubical complexes, which possess a number of properties not shared by all median graphs.
TL;DR: The results show that the hardness of the graphs produced by this method depends in a crucial way on the construction parameters; for a given edge density, challenging graphs can only be constructed using this method for a certain range of maximum clique values.
Abstract: We describe and analyze test case generators for the maximum clique problem (or equivalently for the maximum independent set or vertex cover problems). The generators produce graphs with specified number of vertices and edges, and known maximum clique size. The experimental hardness of the test cases is evaluated in relation to several heuristics for the maximum clique problem, based on neural networks, and derived from the work of A. Jagota. Our results show that the hardness of the graphs produced by this method depends in a crucial way on the construction parameters; for a given edge density, challenging graphs can only be constructed using this method for a certain range of maximum clique values; the location of this range depends on the expected maximum clique size for random graphs of that density; the size of the range depends on the density of the graph. We also show that one of the algorithms, based on reinforcement learning techniques, has more success than the others at solving the test cases p...
CEMA1
TL;DR: It is proved that every graph is an intersection graph of a Helly family of subtrees of a graph without triangles, and polynomial-time recognition algorithms for the intersection graphs and for the perfect intersection graphs of Helly families of subtree in cacti graphs are described.
TL;DR: This proof is conceptually new: it does not use the “replication” operation, or any kind of polyhedral argument; the arguments resemble more the well-known ways of deducing structural properties of minimal imperfect graphs.
Abstract: We provide a very short proof of the following theorem of Lovasz, and of its consequences:A graph is perfect if and only if in every induced subgraph the number of vertices does not exceed the product of the stability and clique numbers of the subgraph.
TL;DR: A linear-time algorithm for finding a minimum r -dominating clique in dually chordal graphs (a generalization of strongly chordal graph) and a simple necessary and sufficient condition for the existence of r -Dominating cliques in the case of Helly graphs and of chordalGraphs.
TL;DR: The matrix completion problem that motivated cycle completable graphs is reviewed and new structural characterizations of these graphs are given, along with their relationship to chordal and series-parallel graphs.
16 Dec 1996
TL;DR: Two variations of the graph searching problem, edge searching and node searching, are studied on several classes of chordal graphs, which include split graphs, interval graphs and k-starlike graphs.
Abstract: Two variations of the graph searching problem, edge searching and node searching, are studied on several classes of chordal graphs, which include split graphs, interval graphs and k-starlike graphs.
TL;DR: If G and all of its powers are chordal then all these graphs admit a common perfect elimination ordering, and such an ordering can be computed in O(| V | · | E |) time using a generalization of the Tarjan and Yannakakis' Maximum Cardinality Search.
12 Jun 1996
TL;DR: This work gives approximate heuristics for MIDS in cubic and at most cubic graphs, based on greedy and local search techniques, for finding an independent dominating set of minimum cardinality in bounded degree and regular graphs.
Abstract: We consider the problem of finding an independent dominating set of minimum cardinality in bounded degree and regular graphs. We first give approximate heuristics for MIDS in cubic and at most cubic graphs, based on greedy and local search techniques.
TL;DR: This work provides an alternate characterization and shows that an O(m + n) recognition algorithm can be derived from the new characterization and constructs the intersection model when the input graph is a circular permutation graph (CPG).
Abstract: An undirected graph G is a circular permutation graph if it can be represented by the following intersection model: Each vertex of G corresponds to a chord in the annular region between two concentric circles, and two vertices are adjacent in G if and only if their corresponding chords intersect each other exactly once. Circular permutation graphs are a generalization of permutation graphs. Rotem and Urrutia introduced and characterized this class of graphs and their characterization yields an O(n 2.376 ) algorithm for recognizing circular permutation graphs. Gardner gave an O(n 2 ) recognition algorithm. We provide an alternate characterization and show that an O(m + n) recognition algorithm can be derived from the new characterization. Our algorithm also constructs the intersection model when the input graph is a circular permutation graph (CPG).