Showing papers on "Chordal graph published in 1981"
TL;DR: It is shown that this family of graphs can be uniquely represented by a tree where the leaves of the tree correspond to the vertices of the graph.
872 citations
TL;DR: The number of complete graphs of order 4 contained in certain graphs is counted to find the total number of graphs in the graph LaSalle's inequality.
323 citations
TL;DR: A family of regular graphs of degree 3, called Chordal Rings, is presented as a possible candidate for the implementation of a local network of message-connected (micro) computers and the diameter of this family is shown to be of 0(n 1/2).
Abstract: A family of regular graphs of degree 3, called Chordal Rings, is presented as a possible candidate for the implementation of a local network of message-connected (micro) computers. For a properly constructed graph in this family having n nodes the diameter, or maximum length message path, is shown to be of 0(n 1/2). The symmetry of the graphs makes it possible to determine message routing by using a simple distributed algorithm. The given algorithm is also potentially useful for the determination of alternate paths in the event of node or link failure.
270 citations
179 citations
TL;DR: The probability that a random labelled r -regular graph contains a given number of cycles of given length is investigated asymptotically and an asymPTotic formula results for the number of labelled r-regular graphs with a given girth.
173 citations
TL;DR: It is shown that the max-cut problem can be solved in polynomial time for weakly bipartite graphs and an algorithm that computes a shortest path of even length is presented.
165 citations
TL;DR: It is proved that the inequalitys≦7 holds for finites-transitive graphs assuming that the list of known 2- transitive permutation groups is complete.
Abstract: We prove that the inequalitys≦7 holds for finites-transitive graphs assuming that the list of known 2-transitive permutation groups is complete.
159 citations
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125 citations
TL;DR: A polynomial time algorithm for testing isomorphism of permutation graphs (comparability graphs of 2-dimensional partial orders) is described and performs two types of simplifying transformations on the graph.
Abstract: A polynomial time algorithm for testing isomorphism of permutation graphs (comparability graphs of 2-dimensional partial orders) is described. It operates by performing two types of simplifying transformations on the graph; the contraction of duplicate vertices and the contraction of uniquely orientable induced subgraphs.
TL;DR: An algorithm based upon Edmonds’s procedure for testing isomorphism of trees is extended to answer various questions concerning automorphisms of a labeled forest.
Abstract: An algorithm based upon Edmonds’s procedure for testing isomorphism of trees is extended to answer various questions concerning automorphisms of a labeled forest. This and linear pattern matching techniques are used to build efficient algorithms which find the automorphism partition and a set of generators for the automorphism group, determine the order of the automorphism group, and compute a coding for forests, interval graphs, outerplanar graphs, and planar graphs.
TL;DR: A new method is developed for constructing graphs with maximum vertex degree 3 and chromatic index 4 and an infinite family of edge-critical graphs with an even number of vertices is constructed, disproves the Critical Graph Conjecture.
TL;DR: It is shown that, for each n, all sufficiently large Paley graphs satisfy Axiom n, which concludes at once that several properties of graphs are not first order, including self-complementarity and regularity.
Abstract: A graph satisfies Axiom n if, for any sequence of 2n of its points, there is another point adjacent to the first n and not to any of the last n. We show that, for each n, all sufficiently large Paley graphs satisfy Axiom n. From this we conclude at once that several properties of graphs are not first order, including self-complementarity and regularity.
TL;DR: A simple linear algorithm is presented for coloring planar graphs with at most five colors using a recursive reduction of a graph involving the deletion of a vertex of degree 6 or less possibly together with the identification of its several neighbors.
TL;DR: Several (nonspectral) classical theorems about line graphs are extended to generalized line graphs, including the derivation and construction of the 31 minimal nongeneralized line graph, a Krausz-type covering characterization, and Whitney-type theorem on isomorphisms and automorphisms.
Abstract: Generalized line graphs extend the ideas of both line graphs and cocktail party graphs. They were originally motivated by spectral considerations. in this paper several (nonspectral) classical theorems about line graphs are extended to generalized line graphs, including the derivation and construction of the 31 minimal nongeneralized line graphs, a Krausz-type covering characterization, and Whitney-type theorems on isomorphisms and automorphisms.
TL;DR: In this paper, a 2-cell imbedding for edge-coloured graphs is introduced, called regular, which is a special case of regular 2-cells imbeddings.
Abstract: A particular kind of 2-cell imbeddings, called regular, for edge-coloured graphs is introduced. By using both geometric and combinatorial techniques, some general imbedding theorems for such graphs, strictly related to the polyhedra they represent, are presented.
TL;DR: There exist infinitely many Kernel-perfect graphs G whose inverse graph G^-^1 has no kernel and there are also infinitely many kernel-critical graphs that are not strongly connected.
TL;DR: Copolychromatic (and therefore codichromatic) graphs of arbitrarily high connectivity are constructed thereby solving a problem posed in Tutte's paper.
TL;DR: Results involving automorphisms and fragments of infinite graphs are proved and it is proved that for a given fragmentC and a vertex-transitive subgroupG of the automorphism group of a connected graph there exists σ≠G such that σ[C] ⊂C.
Abstract: Results involving automorphisms and fragments of infinite graphs are proved. In particular for a given fragmentC and a vertex-transitive subgroupG of the automorphism group of a connected graph there exists σ≠G such that σ[C] ⊂C. This proves the countable case of a conjecture of L. Babai and M. E. Watkins concerning graphs allowing a vertex-transitive torsion group action.
TL;DR: It is shown that edge deletion and edge contraction problems are NP-hard if π is finitely characterizable by 3-connected graphs.
TL;DR: An O(n2K+1) algorithm for the maximum weighted vertex packing problem for graphs inG ≥ 1 and an algorithm for piecing together maximum weighted packings on blocks to find maximum weightedPackings on graphs that contain more than one block are presented.
Abstract: The vertex packing problem for a given graph is to find a maximum number of vertices no two of which are joined by an edge. The weighted version of this problem is to find a vertex packingP such that the sum of the individual vertex weights is maximum. LetG be the family of graphs whose longest odd cycle is of length not greater than 2K + 1, whereK is any non-negative integer independent of the number (denoted byn) of vertices in the graph. We present an O(n
2K+1) algorithm for the maximum weighted vertex packing problem for graphs inG ≥ 1. A by-product of this algorithm is an algorithm for piecing together maximum weighted packings on blocks to find maximum weighted packings on graphs that contain more than one block. We also give an O(n
2K+3) algorithm for testing membership inG
TL;DR: The paper investigates hierarchies of families of graphs obtained by this mechanism, both in the directed and undirected case.
TL;DR: K-regular graphs with specified edge connectivity are considered and some classical theorems and some new results concerning the existence of matchings can be proved by using the polyhedral characterization of Edmonds by showing how lower bounds on the number of perfect matchings in bicritical graphs can be improved.
TL;DR: Techniques are demonstrated by which an asymptotic description of the behavior of a greedy algorithm for this problem is determined, in the case in which the edge weights are drawn from a normal distribution.
Abstract: Optimization problems on complete graphs with edge weights drawn independently, from a fixed distribution, are considered. Several methods for analyzing these problems are discussed, including greedy methods, applications of Boole’s inequality, and exploitation of relationships with results about random unweighted graphs. These techniques are illustrated in the case in which the edge weights are drawn from a normal distribution; in particular, we investigate the expected behavior of the minimum weight clique on k vertices. We describe the asymptotic behavior (in probability and/or almost surely) of the random variable which describes the optimum; we also discuss the asymptotic behavior of its mean. Finally techniques are demonstrated by which we may determine an asymptotic description of the behavior of a greedy algorithm for this problem.
TL;DR: This paper gives a direct proof that (suitably defined) combinatorially homogeneous graphs are ultrahomogeneous.
TL;DR: These include trees, maximal outplanar graphs, k-trees, chordal graphs, and minimally two-connected graphs, which have recursive representations and invariants of recursive labelings of some of these graphs are investigated.
Abstract: We consider classes of undirected, not weighted graphs which have recursive representations; these include trees, maximal outplanar graphs, k-trees, chordal graphs, and minimally two-connected graphs We investigate invariants of recursive labelings of some of these graphs One consequence of the existence of such invariant relation is that we can describe a single-source, shortest-paths spanning tree in terms of the recursive representation We also discuss reasons why we cannot do this as well for other types of recursive graphs
TL;DR: In this paper, a necessary and sufficient condition for a graph H to be the clique graph of some graph G without multicliqual edges was given, and it was shown that there are infinitely many self-clique graphs having more than one critical generator.
Abstract: An edge which belongs to more than one clique of a given graph is called a multicliqual edge. We find a necessary and sufficient condition for a graph H to be the clique graph of some graph G without multicliqual edges. We also give a characterization of graphs without multicliqual edges that have a unique critical generator. Finally, it is shown that there are infinitely many self-clique graphs having more than one critical generator.