Showing papers on "Split graph published in 1987"
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01 Jan 1987
TL;DR: In this paper, the authors introduce the concept of intersection graphs and define a set of classes of intersection graph classes: Parsimonious Set Representations, Clique Graphs Line Graphs Hypergraphs, Split Graphs, Interval Graphs and Threshold Graphs.
Abstract: Preface 1. Intersection Graphs. Basic Concepts Intersection Classes Parsimonious Set Representations Clique Graphs Line Graphs Hypergraphs 2. Chordal Graphs. Chordal Graphs as Intersection Graphs Other Characterizations Tree Hypergraphs Some Applications of Chordal Graphs Split Graphs 3. Interval Graphs. Definitions and Characterizations Interval Hypergraphs Proper Interval Graphs Some Applications of Interval Graphs 4. Competition Graphs. Neighborhood Graphs Competition Graphs Interval Competition Graphs Upper Bound Graphs 5. Threshold Graphs. Definitions and Characterizations Threshold Graphs as Intersection Graphs Difference Graphs and Ferrers Digraphs Some Applications of Threshold Graphs 6. Other Kinds of Intersection. p-Intersection Graphs Intersection Multigraphs and Pseudographs Tolerance Intersection Graphs 7. Guide to Related Topics. Assorted Geometric Intersection Graphs Bipartite Intersection Graphs, Intersection Digraphs, and Catch (Di)Graphs Chordal Bipartite and Weakly Chordal Graphs Circle Graphs and Permutation Graphs Clique Graphs of Chordal Graphs and Clique-Helly Graphs Containment, Comparability, Cocomparability, and Asteroidal Triple-Free Graphs Infinite Intersection Graphs Miscellaneous Topics P4-Free Chordal Graphs and Cographs Powers of Intersection Graphs Sphere-of-Influence Graphs Strongly Chordal Graphs Bibliography Index.
483 citations
TL;DR: Methods of thermodynamical simulation used for combinatorial optimization problems are described and an approach to partition of the node set into as few independent sets as possible is combined with other techniques for graph coloring.
214 citations
TL;DR: This paper determines all weakly symmetric graphs of order twice a prime and shows that these graphs too are directed-edge transitive.
178 citations
TL;DR: A graph is called “perfectly orderable” if its vertices can be ordered in such a way that, for each induced subgraph F, a certain “greedy" coloring heuristic delivers an optimal coloring of F.
Abstract: A graph is called “perfectly orderable” if its vertices can be ordered in such a way that, for each induced subgraph F, a certain “greedy” coloring heuristic delivers an optimal coloring of F. No polynomial-time algorithm to recognize these graphs is known. We present four classes of perfectly orderable graphs: Welsh–Powell perfect graphs, Matula perfect graphs, graphs of Dilworth number at most three, and unions of two threshold graphs. Graphs in each of the first three classes are recognizable in a polynomial time. In every graph that belongs to one of the first two classes, we can find a largest clique and an optimal coloring in a linear time.
137 citations
TL;DR: This work investigates the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic conveXity, in particular, those graphs in which balls around nodes are convex, and those graph in which neighborhoods of convex sets are conveX.
112 citations
TL;DR: A theorem of Moon and Moser is generalized to determine the maximum number of maximal independent sets in a connected graph on n vertices for n sufficiently large, e.g., n > 50.
Abstract: Generalizing a theorem of Moon and Moser, we determine the maximum number of maximal independent sets in a connected graph on n vertices for n sufficiently large, eg, n > 50
102 citations
TL;DR: An algorithm with time bound O ( n 2 ) for the weighted independent domination problem on permutation graphs and an investigation of (weighted) dominating clique problems for several graph classes including an NP-completeness result for weakly triangulated graphs as well as polynomial time bounds are given.
82 citations
TL;DR: This note proves the Strong Perfect Graph Conjecture for ( K 4 − e )-free graphs from first principles and directly yields an O ( pn 2 ) algorithm for p -coloring a perfect ( K4 − e)-free graph.
76 citations
TL;DR: Several new classes of graphs on which the maximum-weight clique problem is solvable in polynomial time are introduced, and the central idea is that every clique of any of the authors' graphs is contained in some member of aPolynomial-sized collection of induced subgraphs that are complements of bipartite graphs.
Abstract: We introduce several new classes of graphs on which the maximum-weight clique problem is solvable in polynomial time. Their common feature, and the central idea of our algorithms, is that every clique of any of our graphs is contained in some member of a polynomial-sized collection of induced subgraphs that are complements of bipartite graphs.
50 citations
TL;DR: This paper presents parallel algorithms for coloring a constant-degree graph with a maximum degree of Δ using Δ + 1 colors and for finding a maximal independent set in a constant,degree graph.
39 citations
01 Jan 1987
TL;DR: An NC algorithm for recognizing chordal graphs, and NC algorithms for finding the following objects on chordal graph graphs: all maximal cliques, an intersection graph representation, an optimal coloring, a perfect elimination scheme, a maximum independent set, a minimum clique cover, and the chromatic polynomial are presented.
Abstract: We present an NC algorithm for recognizing chordal graphs, and we present NC algorithms for finding the following objects on chordal graphs: all maximal cliques, an intersection graph representation, an optimal coloring, a perfect elimination scheme, a maximum independent set, a minimum clique cover, and the chromatic polynomial The well known polynomial algorithms for these problems seem highly sequential, and therefore a different approach is needed to find parallel algorithms
TL;DR: This paper contains several results concerning geodesically convex sets in bridged graphs and obtains two recursive characterizations of the class of bridged graph.
TL;DR: This work improves, for these classes of graphs, Bouchet's 216-flow theorem and approaches his 6-flow conjecture by proving it for a class of 3-connected graphs.
TL;DR: The notion of 1-inseparable graphs is “parallel” to that of biconnected graphs in that different edges in different inseparable components of a graph are not contained in any induced cycle or any complement of an induced cycle.
TL;DR: It is shown that BANDWIDTH can be solved in time O(n2) for interval graphs and that for a given interval graph a linear layout with minimum bandwidth can be constructed in time N2logn.
Abstract: An assignment of unique integers to the vertices of a graph is called a linear layout. The bandwidth minimization problem (BANDWIDTH) is the following: Given a graph G = (V, E) and an integer k, determine whether there exists a linear layout of G such that the maximum difference between adjacent vertices is bounded by k. Interval graphs are the intersection graphs of a family of intervals of the real line. BANDWIDTH remains NP-complete even when restricted to special subclasses of trees. We show that BANDWIDTH can be solved in time O(n2) for interval graphs. Moreover, for a given interval graph a linear layout with minimum bandwidth can be constructed in time O(n2logn). As a by-product we get that this construction can be done for proper interval graphs in time O(nlog n+m).
TL;DR: An algorithm is presented that, given the intersection model S of a circular-arc graph G with n vertices and m edges, finds a maximum-sized clique of G in O(n 2 log log n) time.
TL;DR: An algorithmic proof of the validity of the Strong Perfect Graph Conjecture for graphs whose largest clique is a triangle is presented and a method is presented to contract a perfect graph into a set of smaller perfect graphs that are ( K 4 -e )-free.
TL;DR: This paper proves that bidegreed graphs (graphs whose vertices all have one of two possible degrees) are edge reconstructible and is generalized to show that all graphs that do not have three consecutive integers in their degree sequence are also edge reconstructionible.
Abstract: An edge-deleted subgraph of a graph G is a subgraph obtained from G by the deletion of an edge. The Edge Reconstruction Conjecture asserts that every simple finite graph with four or more edges is determined uniquely, up to isomorphism, by its collection of edge-deleted subgraphs. A class of graphs is said to be edge reconstructible if there is no graph in the class with four or more edges that is not edge reconstructible. This paper proves that bidegreed graphs (graphs whose vertices all have one of two possible degrees) are edge reconstructible. The results are then generalized to show that all graphs that do not have three consecutive integers in their degree sequence are also edge reconstructible.
TL;DR: Some quantitative results are established for the special class of graphs which contain no isometric cycles other than triangles, and it is shown how each cycle in such a graph may be decomposed into chordal pieces.
Abstract: A combinatorial notion of null-homotopy for graphs was introduced by Duchet, Las Vergnas, and Meyniel. Their results were of a qualitative nature for all such graphs. Here some quantitative results are established for the special class of graphs which contain no isometric cycles other than triangles. It is also shown how each cycle in such a graph may be decomposed into chordal pieces.
TL;DR: This note addresses the following question: Which graphs G on n vertices with w ( G ) = r have the maximum number of cliques?
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TL;DR: A theorem of Meyniel about minimal imperfect graphs to partitionable graphs is generalized and follows easily from a theorem of Bland, Huang, and Trotter.
TL;DR: GivenG, a graph, the cochromatic number, Z(G), ofG is the fewest number of sets into which the vertex set can be partitioned so that each set induces a complete or an empty graph.
Abstract: GivenG, a graph, the cochromatic number,Z(G), ofG is the fewest number of sets into which the vertex set can be partitioned so that each set induces a complete or an empty graph. A graph is critically cochromatic if the removal of any of its vertices decreases its cochromatic number. A graph is uniquely cochromatic if there is exactly one partition of minimum order in which each set induces a complete or an empty graph. A graph is comaximal if the removal of any edge increases its cochromatic number. These and related concepts are examined.
TL;DR: Series-parallel graphs, outerplanar graphs, and graphs whose polygon matroids are transversal have been characterized by forbidden subgraphs are related to properties of the graph decomposition.
Abstract: Series-parallel graphs, outerplanar graphs, and graphs whose polygon matroids are transversal have been characterized by forbidden subgraphs. Tutte introduced a graph decomposition for nonseparable graphs. The results of this paper relate the existence of the forbidden subgraphs to properties of the decomposition. Algorithmic implications are considered.
TL;DR: Three different characterizations of intersection graphs of halflines in R 1 are given and the number of such graphs on n vertices is determined and sphericity of joins of triangulated graphs with bipartite complements is proved.
TL;DR: The theory of simplicial decompositions appears to be a very interesting, but still largely unexploited, method of characterization in graph theory, which seems tailor-made for problems like the one discussed.
Abstract: Every planar triangulation G has the property that each induced cycle C of length at least 4 in G separates G, but no proper subgraph of C does. This property is trivially shared by all chordal graphs since these contain no such cycles at all. We ask to what extent maximally planar graphs and chordal graphs are unique with this property — or how much larger the class of graphs is that it determines. The answer is given in the form of a characterization of this class in terms of the simplicial decompositions of its elements. The theory of simplicial decompositions appears to be a very interesting, but still largely unexploited, method of characterization in graph theory, which seems tailor-made for problems like the one discussed.
TL;DR: It is shown that the only graphs for which equality holds in this inequality are the Turin graphs with the same number of vertices in each partite set.
Abstract: An inequality relating the size and order of a simple graph to the average number of triangles containing a fixed edge is proven It is shown that the only graphs for which equality holds in this inequality are the Turin graphs with the same number of vertices in each partite set
TL;DR: It is shown that for many a type, the number of induced subgraphs of that type in G n is asymptotically normally distributed as n tends to infinity.
TL;DR: In this paper, a general formula for the matching polynomial of an arbitrary graph G is derived, which yields a method for counting matchings in graphs, and explicit formulae are deduced for the number of k-matchings in several well-known families of graphs.
Abstract: A general formula is derived for the matching polynomial of an arbitrary graph G. This yields a method for counting matchings in graphs. From the general formula, explicit formulae are deduced for the number of k-matchings in several well-known families of graphs.
TL;DR: This paper presents an algorithm for detection and solution of implicit forms within a model, thereby providing an opportunity for efficient numerical solution, and includes a brief introduction to bond graphs via an electromechanical system example.
Abstract: Bond graphs may be used to model the power flow in dynamic systems. They are especially attractive for modeling systems which function in coupled energy domains, for example, electromechanical systems. For such systems, bond graphs can be used to provide a natural subdivision into power/energy fields: storage, sources, transformers, and dissipation. In the case of nonlinear dissipative fields, implicit, nonlinear, coupled systems of algebraic equations may arise. Causality assignment on the bond graph provides a basis for detecting implicit formulations. This paper presents an algorithm for detection and solution of these forms within a model, thereby providing an opportunity for efficient numerical solution, and includes a brief introduction to bond graphs via an electromechanical system example.