Showing papers on "Connectivity published in 1972"

State Assignment of Asynchronous Sequential Machines Using Graph Techniques

Abstract: A new algorithm for asynchronous machines is presented. The problem is formulated and solved in graph-theoretic terms. A graph-embedding algorithm in an n-cube developed in Sections I-III is used to establish the encoding method. The resulting critical race free encoding minimizes the dimensions of the internal variables and the transition times. This method should be especially interesting for large incomplete machines.

Estimating the connectivity of a graph

Abstract: Although the results above seem to yield good estimators of an upper bound on connectivity, they can easily give rather poor estimates of k itself. One possible improvement could be effected by using the theorem of Harary and Chartrand [2] that δ≥p−2+n / 2 for some n such that 1≤n≤p−1 implies k≧n. This could give an estimate of a lower bound on k by using δ* in place of δ in the above inequality. Of course, the usefulness of this is limited to cases in which δ* is rather large, at least 1/2 p.

Three Results for Trees, Using Mathematical Induction

Abstract: This paper presents three theorems each of which extends to trees a result that is known or obvious for a line segment. While the theorems are of interest in themselves, it is the purpose of the paper to use them as examples of how the inductive process may be effective in proving theorems for trees or for graphs with treelike structures. Other examples of the application of induction to trees include the proof that a tree with n nodes has n 1 arcs (cf. [5],· p. 35), or a paper such as [4]. The definition of a tree which will be used throughout is the followin g: A tree is a connected graph containing no cycles.

Connectivity, Line-Connectivity and J-Connection of the Total Graph

Abstract: Our aim is to prove some results concerning the connectivity, lineconnectivity and J-connection or connection modulo J of total graphs. Only finite, nonoriented graphs without loops or multiple edges will be considered. Let G be a graph whose vertex set is V(G) and whose edge set is E(G). The elements of the set V(G)wE(G) will be called the elements of G and two elements of G are said to be associated if they are either adjacent or incident. The total graph T(G) of G is a graph whose vertex set is V(G)uE(G), two vertices being joined by an edge if and only if they are associated elements of G (see [1]). As an example of a graph G and its total graph T(G) see Fig. 1. We make an obvious distinction: small rings represent point-vertices of T(G) (x is a point-vertex x ~ E(G)). T(G) contains both G and its line-graph or interchange graph [4] L(G) as disjoint subgraphs. Remember that by definition, V[L(G)] =E(G) and two vertices of L(G) are linked by an edge if and only if the corresponding edges of G are adjacent. Edges of T(G) belonging neither to G nor L(G) form what will be called the incidence-graph I(G) of G. By definition the connectivity k(G) of G is the least number of vertices whose removal disconnects G or reduces G to a single vertex; a set of k(G) vertices satisfying this condition is called a minimal separating vertex set of G. Moreover G is n-connected if and only if k(G) >= n. On the