Showing papers on "Connectivity published in 1974"

On the connectivity of social networks

Abstract: In the context of analyzing social structures through the use of graph theory, some binary matrix operations for valued graphs are presented. These operations are demonstrated in the analysis of social network data presented by Kapferer. Further examples of where the procedures should prove useful are suggested.

On covering the points of a graph with point disjoint paths

Abstract: The minimum number of point disjoint paths which cover all the points of a graph defines a covering number denoted by ζ. The relation of ζ to some other well-known graphical invariants is discussed, and ζ is evaluated for a variety of special classes of graphs. A simple algorithm is developed for determining ζ in the case of a tree, and it is shown that this tree algorithm can be generalized to yield ζ for any connected graph. Degree conditions are also derived which yield simple upper bounds for ζ.

On the hamiltonian completion problem

Abstract: We define the Hamiltonian completion number of a graph G, denoted hc(G), to be the minimum number of lines that need to be added to G in order to make it Hamiltonian. The Hamiltonian completion problem asks for hc(G) and a specific Hamiltonian cycle containing hc(G) new lines. We derive an efficient algorithm for finding hc(T) for any tree T, and show that if S is the set of spanning trees of an arbitrary connected graph G, then Open image in new window .

Testing graph connectivity

Abstract: An algorithm proposed by Dinic for finding maximum flows in networks and by Hopcroft and Karp for finding maximum bipartite matchings is applied to graph connectivity problems. It is shown that the algorithm requires 0(V1/2E) time to find a maximum set of node-disjoint paths in a graph, and 0(V2/3E) time to find a maximum set of edge disjoint paths. These bounds are tight. Thus the node connectivity of a graph may be tested in 0(V5/2E) time, and the edge connectivity of a graph may be tested in 0(V5/3E) time.

The genera of edge amalgamations of complete bigraphs

Abstract: If G and H are graphs, then G V H is defined to be a graph obtained by identifying some edge of G with some edge of H. It is shown that for all m, n, p, and q the genus g(K nV Kp ) is either g(K n) + g(K PI or else g(K m,n) + g(Kp ) 1. The latter value is attained if and only if both K and K are critical in the sense that the deletion of any edge results m,n P,q in a graph whose genus is one less than the genus of the original graph. I. Throughout this paper graphs will be finite simplicial 1-complexes. The genus g(G) of a connected graph G is the minimum genus of any closed orientable 2-manifold in which G can be imbedded. If T is a subgraph both of G and of H, then a new graph G VT H may be formed by identifying a copy of T contained in G with a copy of T contained in H. This new graph is called an amalgamation of G and H along T, and in general depends on the choice of copies of T. The aim of this paper is to determine the genera of all graphs of the form Km,n VK2 KP q. This is achieved in Theorem I.3. Note that, due to the symmetry of complete bigraphs, the amalgamation Km n VK2 KP q is independent of which edges K2 one amalgamates along. To simplify notation, an amalgamation G VK2 H shall be written simply as G V H. Using different methods, Ringel [3] and Schanuel [4] have determined the genera of all complete bigraphs, as follows. Theorem I. 1. For all integers m and n greater than or equal to 2, g(Km,n) = (m 2)(n -2)/4, where lrt denotes the least integer greater than or equal to any real number r. Received by the editors December 21, 1972. AAIS (AIOS) subject classifications (1970). Primary 05C1b; Secondary 55A15.

CHAPTER 4 – Directed Graphs

Abstract: Publisher Summary This chapter focuses on directed graphs and associated concepts, and examines some of the uses of these graphs. In the study of directed graphs, multiple edges are often quite significant. Multiple arcs joining two vertices can be either parallel, that is, directed in the same direction; or antiparallel, that is, directed in opposite direction. As the directions on the arcs are used to indicate a flow of information or material, it is important to permit the use of multiple arcs in directed graphs. In a directed graph, a loop is an arc beginning and terminating at the same vertex. A directed graph is weakly connected if the same graph as an undirected graph, that is, ignoring the directions of the arcs, is a connected graph. A directed graph is unilaterally connected if, given any two vertices of the graph, there exists a path from one vertex to the other, although a reverse path does not necessarily exist. It is possible to split the degree of a vertex into two numbers called the indegree, which is the number of arcs incident to a vertex, and the outdegree, which is the number of arcs incident from a vertex.