Showing papers on "Connectivity published in 1984"
TL;DR: In this article, an analogous difference operator is studied for an arbitrary graph, and it is shown that many properties of the Laplacian in the continuous setting (e.g., the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator.
Abstract: The difference Laplacian on a square lattice in Rn has been stud- ied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference oper- ator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient. The random walks we consider are defined as follows. Let K be a connected graph (i.e. a one dimensional simplicial complex). For a vertex x E K, let m(x) denote the number of edges emanating from x. The probability that a particle moves from x to another vertex y E K is l/m(x) if x and y are connected by an edge and it is zero otherwise. As observed by Courant, Friedrichs and Lewy (CFL) for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the Laplacian
527 citations
TL;DR: The purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing AND’'s, OR’s and NOTs, with no bound placed on the fan-in of AND-gates and OR-Gates.
Abstract: The purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing AND’s, OR’s and NOT’s, with no bound placed on the fan-in of AND-gates and OR-gates. Two such reducibilities are defined, and reductions and equivalences among several common problems such as parity, sorting, integer multiplication, graph connectivity, bipartite matching and network flow are given. Certain problems are shown to be complete, with respect to these reducibilities, in the complexity classes deterministic logarithmic space, nondeterministic logarithmic space, and deterministic polynomial time. New upper bounds on the size-depth (unbounded fan-in) circuit complexity of symmetric Boolean functions are established.
341 citations
Book•
01 Sep 1984
TL;DR: In this article, the authors present a survey of algorithms on graphs and their applications in the literature, including the following: 1. Graph Algorithms and NP-Completeness, 2. Topological sorting and the Representation Problem, 3. Transitive Closure of Acyclic Digraphs, 4. General Path Problems and Matrix Multiplication, and 5. More NP-complete Problems.
Abstract: Vol. 2: Graph Algorithms and NP-Completeness.- IV. Algorithms on Graphs.- 1. Graphs and their Representation in a Computer.- 2. Topological Sorting and the Representation Problem.- 3. Transitive Closure of Acyclic Digraphs.- 4. Systematic Exploration of a Graph.- 5. A Close Look at Depth First Search.- 6. Strongly-Connected and Biconnected Components of Directed and Undirected Graphs.- 7. Least Cost Paths in Networks.- 7.1. Acyclic Networks.- 7.2. Non-negative Networks.- 7.3. General Networks.- 7.4. The All Pairs Problem.- 8. Minimum Spanning Trees.- 9. Maximum Network Flow and Applications.- 9.1 Algorithms for Maximum Network Flow.- 9.2 (0,1)-Networks, Bipartite Matching and Graph Connectivity.- 9.3 Weighted Network Flow and Weighted Bipartite Matching.- 10. Planar Graphs.- 11. Exercises.- 12. Bibliographic Notes.- V. Path Problems in Graphs and Matrix Multiplication.- 1. General Path Problems.- 2. Two Special Cases: Least Cost Paths and Transitive Closure.- 3. General Path Problems and Matrix Multiplication.- 4. Matrix Multiplication in a Ring.- 5. Boolean Matrix Multiplication and Transitive Closure.- 6. (Min,+)-Product of Matrices and Least Cost Paths.- 7. A Lower Bound on the Monotone Complexity of Matrix Multiplication.- 8. Exercises.- 9. Bibliographic Notes.- VI. NP-Completeness.- 1. Turing Machines and Random Access Machines.- 2. Problems, Languages and Optimization Problems.- 3. Reductions and NP-complete Problems.- 4. The Satisfiability Problem is NP-complete.- 5. More NP-complete Problems.- 6. Solving NP-complete Problems.- 6.1 Dynamic Programming.- 6.2 Branch and Bound.- 7. Approximation Algorithms.- 7.1 Approximation Algorithms for the Travelling Salesman Problem.- 7.2 Approximation Schemes.- 7.3 Full Approximation Schemes.- 8. The Landscape of Complexity Classes.- 9. Exercises.- 10. Bibliographic Notes.- IX. Algorithmic Paradigms.
157 citations
TL;DR: The speed-up of this algorithm is optimal in the sense that the depth of the algorithm is of the order of the running time of the fastest known sequential algorithm over the number of processors used.
47 citations
TL;DR: In this article, the equivalence between blocks and biconnected components (subgraphs in which there are two edge-disjoint paths between any pair of nodes) that holds in ordinary graph theory can be generalized to hypergraphs.
29 citations
TL;DR: Parallel Breadth-First Search (BFS) algorithms for ordered trees and graphs on a shared memory model of a Single Instruction-stream Multiple Data-stream computer are proposed.
Abstract: Parallel Breadth-First Search (BFS) algorithms for ordered trees and graphs on a shared memory model of a Single Instruction-stream Multiple Data-stream computer are proposed. The parallel BFS algorithm for trees computes the BFS rank of eachnode of an ordered tree consisting of n nodes in time of 0(β log n) when 0(n 1+1/β) processors are used, β being an integer greater than or equal to 2. The parallel BFS algorithm for graphs produces Breadth-First Spanning Trees (BFSTs) of a directedgraph G having n nodes in time 0(log d.log n) using 0(n 3) processors, where d is the diameter of G If G is a strongly connected graph or a connected undirected graph the BFS algorithm produces n BFSTs, each BFST having a different start node.
27 citations
TL;DR: It is proved that the maximum number of vertices of degree 1 of a spanning tree of G is bounded by 14n and 12(n - 2).
21 citations
TL;DR: It is shown that an array of n + 1 cells can be used for a graph with n vertices to find the connected components, a spanning tree, or, when used in conjunction with a systolic priority queue, a minimum spanning tree.
Abstract: In this paper we present a design, suited to VLSI implementation, for a one-dimensional array to solve graph connectivity problems. The computational model is relatively primitive in that only the two end cells of the array can interact with the external environment and only adjacent cells in the array are allowed to communicate. However, we show that an array of n + 1 cells can be used for a graph with n vertices to find the connected components, a spanning tree, or, when used in conjunction with a systolic priority queue, a minimum spanning tree. The area, time, and I/O requirements compare favorably with other models proposed for this problem in the case of sparse graphs.
16 citations
TL;DR: In this article, it was shown that the crossing number of graphs with connectivity 2 has an additive property analogous to that of graphs having connectivity ≤ 1, which is called additive crossing number 2.
Abstract: We prove that the crossing number of graphs with connectivity 2 has in certain cases an additive property analogous to that of crossing number of graphs with connectivity ≤1.
11 citations
03 Sep 1984
7 citations
01 Jul 1984
TL;DR: Three methods of augmenting computer networks by adding at most one link per processor are discussed.
Abstract: Three methods of augmenting computer networks by adding at most one link per processor are discussed: (1) A tree of N nodes may be augmented such that the resulting graph has diameter no greater than 4log sub 2((N+2)/3)-2. Thi O(N(3)) algorithm can be applied to any spanning tree of a connected graph to reduce the diameter of that graph to O(log N); (2) Given a binary tree T and a chain C of N nodes each, C may be augmented to produce C so that T is a subgraph of C. This algorithm is O(N) and may be used to produce augmented chains or rings that have diameter no greater than 2log sub 2((N+2)/3) and are planar; (3) Any rectangular two-dimensional 4 (8) nearest neighbor array of size N = 2(k) may be augmented so that it can emulate a single step shuffle-exchange network of size N/2 in 3(t) time steps.
TL;DR: At the 4th International Graph Theory Conference 1980, G. Chartrand posed the following problem: If a (connected) graph G contains spanning trees with m and n pendant vertices, respectively, with m < n, does G contain a spanning tree with k pendant Vertices for every integer k, where m < k < n?
TL;DR: A new algorithm to determine all minimal cuts up to third order that isolate some sink node from all source nodes in a planar graph that has the advantage of having a linear complexity, which makes the problem tractable as opposed to path oriented methods, where path determination grows exponentially with the size of the graph.
Abstract: This paper presents a new algorithm to determine all minimal cuts up to third order that isolate some sink node from all source nodes in a planar graph. The algorithm has the advantage of having a linear complexity, which makes the problem tractable as opposed to path oriented methods, where path determination grows exponentially with the size of the graph. This algorithm can be used when the size of the graph requires computer assistance, and it can simplify the application to large systems, of reliability evaluation techniques based on minimal cuts. The limitation of cuts up to third order has a numerical reason since cuts of higher order often negligibly affect the system indexes. A computer application to a graph that models an urban power distribution network shows the algorithm's capacity to handle complex problems and reduce CPU time.
01 Jan 1984
TL;DR: In this paper, it was shown that if a connected graph G contains two distinct spanning trees, then any twospanning trees of G can be connected by a chain of spanning trees.
Abstract: It is proved that if a connected graph G contains two distinct spanning trees,then any twospanning trees of G can be connected by a chain of spanning trees,in which any two consecutivetrees T_4 and T_(i+1) are adjacent,i.e.,the symmetric differene E(T_4)ΔE(T_(i+1)) consists of two adjacentedges.
TL;DR: The conjecture that if G = (V, E) is a connected graph with all valencies ≥k and a1,…,ak ≥ 2 are integers with Σ ai = |V |, then V may be decomposed into subsets A1,….,Ak so that |Ai | = ai and the subgraph spanned by Ai in G has no isolated vertices.
01 Jan 1984
TL;DR: The distance distribution of a connected graph of diameter k is (D1,D2,...,Dk), where Di is the number of pairs of vertices at distance i from one another.
Abstract: The distance distribution (dd) of a connected graph of diameter k is (D1,D2,...,Dk), where Di is the number of pairs of vertices at distance i from one another. The common neighbor distribution (nd) is (n0,n1,n2,...,nn–2), where ni is the number of pairs of vertices having i common neighbors. These and other sequences have been introduced recently as tools in distinguishing pairs of nonisomorphic graphs (dd(G) works best for graphs of large diameter; whereas, nd(G) is more useful for graphs of small diameter). They have also been used to study structural similarity in graphs sharing a common sequence.
01 Jan 1984
TL;DR: For setting up a theory of flows and tensions the concepts of cycle and cocycle are of fundamental significance and should be put at the top of the considerations.
Abstract: For setting up a theory of flows and tensions the concepts of cycle and cocycle are of fundamental significance. Therefore, we want to put them at the top of our considerations.
TL;DR: In this article, a necessary and sufficient condition for H to be a matroid is that, for any weight function W defined on S, the algorithm of Burns and Haff gives a labelling of the family of maximal sets in I as B"1, B"2,..., B'n such that W(B"1) =< W(b"2) => B"n) = > W(w(b) n) = < W(n) n>