Showing papers on "Connectivity published in 1986"

An algorithm for the Rural Postman problem on a directed graph

Abstract: The Directed Rural Postman Problem (DRPP) is a general case of the Chinese Postman Problem where a subset of the set of arcs of a given directed graph is ‘required’ to be traversed at minimum cost. If this subset does not form a weakly connected graph but forms a number of disconnected components the problem is NP-Complete, and is also a generalization of the asymmetric Travelling Salesman Problem. In this paper we present a branch and bound algorithm for the exact solution of the DRPP based on bounds computed from Lagrangean Relaxation (with shortest spanning arborescence sub-problems) and on the fathoming of some of the tree nodes by the solution of minimum cost flow problems. Computational results are given for graphs of up to 80 vertices, 179 arcs and 71 ‘required’ arcs.

On unreliability polynomials and graph connectivity in reliable network synthesis

Abstract: The analysis and synthesis of reliable large-scale networks typically involve a graph theoretic model. We give a survey of the graph theoretic notions which are relevant to the synthesis problem. It is shown how a number of unsolved graph extremal problems relate to the synthesis question.

Connected domatic number of a graph

Abstract: All graphs considered in this paper are finite graphs without loops and multiple edges. The domatic number of a graph was defined by E. J. Cockayne and S. T. Hedetniemi [1]. Later some related concepts were introduced. The same authors together with R. M. Dawes [2] have introduced the total domatic number; R. Laskar and S. T. Hedetniemi [3] have introduced the connected domatic number. A dominating set (or a total dominating set) in an undirected graph G is a subset D of the vertex set V(G) of G with the property that to each vertex x e V(G) — D (or to each vertex xeV(G) respectively) there exists a vertex y e D adjacent to x. A connected dominating set of G is a dominating set of G with the property that the subgraph of G induced by it is connected. A domatic (or total domatic, or connected domatic) partition of G is a partition of V(G), all of whose classes are dominating (or total dominating, or connected dominating, respectively) sets of G. The maximum number of classes of a domatic (or total domatic, or connected domatic) partition of G is called the domatic (or total domatic, or connected domatic, respectively) number of G. The domatic number of G is denoted by d(G), its total domatic number by <4(G), its connected domatic number by dc(G). The connected domatic number of a graph is well defined only for connected graphs; in a disconnected graph there exists no connected dominating set and thus no connected domatic partition, while in every connected graph there exists at least one connected domatic partition, namely that which consists of one class. The connected domatic number of G is closely related to the vertex connectivity number of G. If G is a connected graph, then a vertex cut of G is a subset R of V(G) with the property that the subgraph of G induced by V(G) — R is disconnected. If G is not a complete graph, then the vertex connectivity number x(G) is the minimum cardinality of a vertex cut of G. If G is a complete graph (i. e. without vertex cuts) with n vertices, then we put x(G) = n — 1. Lemma. Let G be a connected graph which is not complete, let R be its vertex cut, let D be its connected dominating set. Then DnR^0.

A physical interpretation of graph connectivity, and its algorithmic applications

Abstract: Connectivity is a basic property of graphs, and is related to other important concepts like reliability, communication and flow. Connectivity is also one of the most well studied areas in graph theory. In this paper, we propose a new point of view on graph connectivity, based on geometric and physical intuition. Our main theorem is a geometric characterization of k-vertex connected graphs. It says that a graph G is k-cQnnected if and only if G has a certain "nondegenerate convex embedding" in (Rk-l. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium. Algebraic properties of this equilibrium ensure that the embedding it defines is nondegenerate exactly when the graph is k-connected. As an application of our theorem we give probabilistic algorithms for computing the connectivity of a graph. The first is a Monte Carlo algorithm that runs in time O(n2.5+nk2.5) where n is the number of vertices and k is the vertex connectivity of the input graph. The second is a Las Vegas algorithm (Le., never errs) that runs in expected time O(kn2.5+nk3.5). For comparison, the best known algorithm (which is deterministic!) runs in time k3nl.5+k2n2 [Gl. Observe that our algorithms are faster for all k ~ In, and for very dens grphs the Monte Carlo algorithm is faster by 8 linear factor! Section 1 contains the main theorem and its proof. In Section 2 we describe the algorithmic applications. These include, in addition to the algorithms above, processor-efficient parallel counte~parts.

A note on chromatic uniqueness of graphs

Abstract: We proved that if a connected graph is chromatically unique, then it has at most two blocks. Furthermore each of these blocks is vertex-transitive and chromatically unique. It is then shown that if the two blocks of a connected graph G are H and K2, then G is chromatically unique if and only if H is vertex-transitive and chromatically unique. This answers a conjecture of Whitehead and Zhao in the affirmative.

Connectivity and chromatic number of infinite graphs

Abstract: Every graph with uncountable chromatic number contains for every finiten ann-connected subgraph with infinite degrees which has uncountable chromatic number.

A lower bound for the convexity number of some graphs

Abstract: Given a connected graph G, we say that a set C ∈ V(G) is convex in G if, for every pair of vertices x, y ∈ C, the vertex set of every x - y geodesic in G is contained in C. The convexity number of G is the cardinality of a maximal proper convex set in G. In this paper, we show that every pair k, n of integers with 2 ≤ k ≤ n - 1 is realizable as the convexity number and order, respectively, of some connected triangle-free graph, and give a lower bound for the convexity number of k-regular graphs of order n with n > k + 1.

Connected graphs containing a given connected graph as a unique greatest common subgraph

Abstract: Those connected graphsG are determined for which there exist nonisomorphic connected graphs of equal size containingG as a unique greatest common subgraph. Analogous results are also obtained for weakly connected and strongly connected digraphs, as well as for induced subgraphs and induced subdigraphs.

A linear programming approach to the optimum network orientation problem

Abstract: In this paper a procedure is described to orient arcs of a graph so as to mimimize the sum of the distances between certain given source-sink pairs. This work is a substantial part of the more general problem of orienting a road network in such a way as to optimize a specified objective subject to the requirement that one can go from any point of the network to any other one. The proposed method, related to the ideas of sequential optimization [1, 2, 3], uses column-generation as well as row-generation.

Review of recent results about theκ-centrum of a tree

Abstract: This paper is concerned with theκ-centrum of a graph This concept, related to a particular location problem, generalizes that of the center and that of the median of a graph: theκ-centrum is the set of points for which the sum of the (weighted) distances from theκ farthest vertices is minimized The paper will review some recent results about this problem In particular, some properties of cardinality, connectivity and, more generally, of the structure of theκ-centrum of a weighted tree will be presented