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Showing papers on "Connectivity published in 1985"


Journal ArticleDOI
TL;DR: This paper clarifies the relation between the diameter k and the edge or node connectivity Ce or c,, of digraphs by derived the following two inequalities: where n is the number of nodes, d is the maximum degree, andd is the minimum degree.
Abstract: This paper clarifies the relation between the diameter k and the edge or node connectivity Ce or c,, of digraphs. The following two inequalities are derived: where n is the number of nodes, d is the maximum degree, and d is the minimum degree.

135 citations


Journal ArticleDOI
TL;DR: In this paper, the minimum diameter, maximum connectivity circulant problem is considered and several results are given for the general case and a simple solution is derived for the connectivity four case.
Abstract: It is well known that maximum connectivity graphs play an important role in the design of reliable networks. The class of symmetric graphs called circulants is known to contain such maximum connectivity graphs. Although not all circulants have this maximum connectivity property, those that do have a great variation in their diameters. Since diameter is a measure of transmission delay, the minimum diameter, maximum connectivity circulant problem is considered here. Several results are given for the general case and a simple solution is derived for the connectivity four case.

116 citations


Proceedings ArticleDOI
21 Oct 1985
TL;DR: A distributed algorithm is presented that constructs the minimum-weight spanning tree of an undirected connected graph with distinct edge weights and distinct node identities with time complexity O(nG(n)+ time units, an improvement from Gallager's O(nlogn)+.
Abstract: A distributed algorithm is presented that constructs the minimum-weight spanning tree of an undirected connected graph with distinct edge weights and distinct node identities. Initially each node knows only the weight of each of its adjacent edges. When the algorithm terminates, each node knows which of its adjacent edges are edges of the tree. For a graph with n nodes and e edges, the total number of messages required by our algorithm is at most 5nlogn+2e, and each message contains at most one edge weight or one node identity plus 3+logn bits. Although our algorithm has the same message complexity as the previously known algorithm by Gallager et al., the time complexity of our algorithm takes at most O(nG(n))+ time units, an improvement from Gallager's O(nlogn)+. A worst case O(nG(n)) is also possible.

90 citations


Journal ArticleDOI
TL;DR: Sabidussi gives a non-algorithmic proof that the cartesian factorization is unique by using a tower of successively coarser equivalence relations on the edge set in which each prime factor of the graph is identified with an equivalence class in the coarsest of the relations.

58 citations


Journal ArticleDOI
TL;DR: In this article, the authors presented an algorithm to find the minimum 3-cut for a planar graph G in O( |V |^2 ) time, where V is the number of vertices in the graph.
Abstract: A 3-cut for a connected graph G is a set of edges which, when deleted, separate G into 3 components. In this paper we present an $O ( | V |^2 )$ algorithm to find the minimum 3-cut for a planar graph G.

33 citations


01 Jan 1985
TL;DR: These algorithms cost only logarithmic time and are the first known that are optimal: the product of their time and processor bounds are bounded by a linear function of the input size.
Abstract: : This document gives new parallel algorithms for integer sorting and undirected graph connectivity problems such as connected components and spanning forest. These algorithms cost only logarithmic time and are the first known that are optimal: the product of their time and processor bounds are bounded by a linear function of the input size. All previous known parallel algorithms for these problems required at least a linear number of processors to achieve logarithmic time bounds, and hence were nonoptimal by at least a logarithmic factor. The author assumes a parallel random access machine (RAM) model which allows both concurrent writes and concurrent reads of global memory. The algorithms are randomized; each processor is allowed an independent random number generator; however our stated resource bounds hold for worst case input with overwhelming likelihood as the input size grows. (Author)

32 citations


Journal ArticleDOI
TL;DR: The Ramsey numberr(F, G) is determined in the case whereF is an arbitrary fixed graph andG is a sufficiently large sparse connected graph with a restriction on the maximum degree of its vertices.
Abstract: The Ramsey numberr(F, G) is determined in the case whereF is an arbitrary fixed graph andG is a sufficiently large sparse connected graph with a restriction on the maximum degree of its vertices. An asymptotically correct upper bound is obtained forr(F, T) whereT is a sufficiently large, but otherwise arbitrary, tree.

26 citations


Journal ArticleDOI
01 Dec 1985-Networks
TL;DR: An extended version of Goldman's algorithm is introduced which (in linear time) either solves (M) on G, or finds the single block of G which contains all solutions to (M).
Abstract: The w-centroid problem, denoted by (C), is an optimization problem which has been shown by Kariv and Hakimi to be equivalent, on a tree graph, to the 1-median location problem, denoted by (M). For a general (weighted) connected graph G we develop a duality between (C) (which is defined on G) and a block optimization problem, denoted by (B), and defined over the blocks of G. A block is a maximal nonseparable subgraph. We analyze (B) and (C) by means of two problems equivalent to (B) and (C) respectively, but defined on a blocking graph G which is always a tree. We give an O(∣V∣) algorithm to solve the two problems on G, and we characterize the solutions. We also show that the solution to a 1-median problem defined on G either solves (M) on the original graph G or localizes the search for a solution to (M) to the vertices of a single block. We introduce an extended version of Goldman's algorithm which (in linear time) either solves (M) on G, or finds the single block of G which contains all solutions to (M).

22 citations


Journal ArticleDOI
TL;DR: It is shown that if the vertices of a Kr are removed from a graph H, then at most r components of the resulting graph contain median Vertices of H, and that graphs having both median and center prescribed are constructed.
Abstract: The distance of a vertex u in a connected graph H is the sum of all the distances from u to the other vertices of H. The median M(H) of H is the subgraph of H induced by the vertices of minimum distance. For any graph G, let f(G) denote the minimum order of a connected graph H satisfying M(H) ≅ G. It is shown that if G has n vertices and minimum degree δ then f(G) ⩽ 2n − δ + 1. Graphs having both median and center prescribed are constructed. It is also shown that if the vertices of a Kr are removed from a graph H, then at most r components of the resulting graph contain median vertices of H.

15 citations


Journal ArticleDOI
TL;DR: It is shown how tree structures can be embedded in a one-dimensional systolic array to solve a connectivity problem, the UNION-FIND problem, by a single left-to-right pass of the data through the array.
Abstract: We show how tree structures can be embedded in a one-dimensional systolic array to solve a connectivity problem, the UNION-FIND problem, by a single left-to-right pass of the data through the array. A previous solution, which did not use trees, required a left-to-right pass followed by a right-to-left pass through the array, as well as a more complex program for each cell.

9 citations


Journal ArticleDOI
TL;DR: Whitney's 2-isomorphism theorem characterizing all graphs with the same cycle matroid and Tutte's excluded minor characterization of those binary matroids that are graphic are given.

Journal ArticleDOI
TL;DR: A new and simpler proof of Hartman's theorem that the cycles of length at least d + 1 generate the cycle space of G, unless d is odd and G ≅ K d +1 .

Journal ArticleDOI
TL;DR: It is proved that there exists a finite graph whose automorphism group has exactly ν orbits on the sets of vertices and ϵ orbit on the set of edges if and only if ν ⩽ 2 ϵ + 1 (resp. a finite connected graph).

01 Jan 1985
TL;DR: This design system takes this graph as input and translates it to an architecture and a control sequence which together realize the functional definition, and has enabled us to build a flexible design system which facilitates a thorough search of the design space.
Abstract: Translating a behavioral description of a digital system into an architecture is the initial step in silicon compilation The behavioral description is in a high level programming language which is converted to a cyclic directed graph, called a data flow graph Our design system takes this graph as input and translates it to an architecture and a control sequence which together realize the functional definition The first step in the translation process is the assignment of priority levels to the nodes of the graph These nodes represent (amongst other things) operations in the behavioral description All nodes belonging to the same priority level are executed at the same instant The second step involves binding nodes of the graph to components of hardware such as operational units, registers and buses through interaction with a hardware data base In the third step (optimization step), the user discards components from the hardware and the system re-sequences the graph to allow for the discarded hardware This approach has enabled us to build a flexible design system which facilitates a thorough search of the design space

01 Jan 1985
TL;DR: In this paper, the formules de recurrence for the probabilite de connectivite de quelques graphes aleatoires dans le cas general ou λab¬=λaλb
Abstract: On etablit les formules de recurrence pour la probabilite de connectivite de quelques graphes aleatoires dans le cas general ou λab¬=λaλb

Journal ArticleDOI
TL;DR: In this article, the authors characterize bipartite randomly near-traceable graphs and show that for every randomly near traceable graph G that is not a cycle, the radius of G is at most 2.
Abstract: A walk generated by a (not necessarily completed) depth-first search of a graph is called a DFS walk. A connected graph is randomly near-traceable if it admits no DFS walk $W:w_1 ,w_2 , \cdots ,w_n $ having consecutive vertices $w_k $ and $w_{k + 1}$ that both appear on the subwalk $w_1 ,w_2 , \cdots ,w_{k - 1} $; thus, in a depth-first search of a randomly near-traceable graph, whenever we backtrack to a previously visited vertex, that vertex is adjacent to at least one unvisited vertex. We characterize the bipartite randomly near-traceable graphs and show that for every randomly near-traceable graph G that is not a cycle, the radius of G is at most 2. Other results are also presented.

Journal ArticleDOI
TL;DR: Two classes of weak contractions are introduced, called inner and outer maps, respectively, which arise naturally in connection with Halin's notion of accessibility, which are closely related to what the authors shall call saturated maps which are characterized by the fact that every edge of the image graph has 'enough' edges in its preimage.