Showing papers on "Connectivity published in 1989"
TL;DR: It is shown that if a connected graph G has minimum degree two and is not one of seven exceptional graphs, then γ(G)γ 2/5|V|, which is the minimum cardinality of a subset of V such that every vertex is either in the set or is adjacent to some vertex in theSet.
Abstract: The domination number γ(G) of a graph G = (V, E) is the minimum cardinality of a subset of V such that every vertex is either in the set or is adjacent to some vertex in the set. We show that if a connected graph G has minimum degree two and is not one of seven exceptional graphs, then γ(G)γ 2/5|V|. We also characterize those connected graphs with γ(G)γ 2/5|V|.
157 citations
TL;DR: Asymptotically sharp upper bounds for the maximum diameter and radius of a connected graph, a connected trangle-free graph, and a connected C 4 -free graph with n vertices and with minimum degree δ are given.
116 citations
TL;DR: It is proved that every connected graph with maximum degree ⩽3 and on more than 7 points has a matching cutset and line graphs with a matchingcutset can be recognized in O(|E|) time.
Abstract: Let G = (V,E) be an undirected graph. A subset F of E is a matching cutset of G if no two edges of F are incident with the same point, and G-F has more components than G. Chvatal [2] proved that it is NP-complete to recognize graphs with a matching cutset even if the input is restricted to graphs with maximum degree 4. We prove the following: (a) Every connected graph with maximum degree ⩽3 and on more than 7 points has a matching cutset. (In particular, there are precisely two connected cubic graphs without a matching cutset). (b) Line graphs with a matching cutset can be recognized in O(|E|) time. (c) Graphs without a chordless circuit of length 5 or more that have a matching cutset can be recognized in O(|V||E|3) time.
57 citations
01 Jan 1989
TL;DR: Tight bounds are determined on the chromatic sum of a connected graph with e edges, which is the smallest possible total among all proper colorings of G using natural numbers.
Abstract: The chromatic sum of a graph is introduced in the dissertation of Ewa Kubicka. It is the smallest possible total among all proper colorings of G using natural numbers. In this article we determine tight bounds on the chromatic sum of a connected graph with e edges,
45 citations
TL;DR: In this article, tight bounds on the chromatic sum of a connected graph with e edges were derived for the case of connected graphs with e vertices, where e is the number of vertices in the graph.
Abstract: The chromatic sum of a graph is introduced in the dissertation of Ewa Kubicka. It is the smallest possible total among all proper colorings of G using natural numbers. In this article we determine tight bounds on the chromatic sum of a connected graph with e edges.
40 citations
TL;DR: In this paper, it was shown that if M is a 3-connected matroid and C is a circuit of M such that M⧹e is not 3connected for every e ∈ C then C meets at least two triads of M.
32 citations
16 Oct 1989
TL;DR: It is shown that, by choosing a multilayer network in such a way that the action of the group on the input nodes can be extended to the whole network, the invariance of the output under the actionof the group can be guaranteed.
Abstract: One of the central tools developed by M. Minsky and S. Papert (1988) was the group invariance theorem. This theorem is concerned with choosing perceptron weights to recognise a predicate that is invariant under a group of permutations of the input. The theorem states that the weights can be chosen to be constant for equivalence classes of predicates under the action of the group. This paper presents this result in a graph theoretic light and then extends consideration to multilayer perceptrons. It is shown that, by choosing a multilayer network in such a way that the action of the group on the input nodes can be extended to the whole network, the invariance of the output under the action of the group can be guaranteed. This greatly reduces the number of degrees of freedom in the training of such a network. An example of using this technique to train a network to recognise isomorphism classes of graphs is given. This compares favourably with previous experiments using standard back-propagation. The connections between the group of symmetries and the network structure are explored and the relation to the problem of graph isomorphism is discussed
31 citations
TL;DR: The edge exchange proofs can be divided into three types, in accordance with the extent to which the exchange sequence depends upon properties of spanning trees, to obtain new interpolation results for some invariants.
Abstract: We say that a graphical invariant i of a graph interpolates over a family F of graphs if i satisfies the following property: If m and M are the minimum and maximum values (respectively) of i over all graphs in F then for each k, m ⩽ k ⩽ M, there is a graph H in F for which i(H)= k. In previous works it was shown that when F is the set of spanning trees of a connected graph G, a large number of invariants interpolate (some of these invariants require the additional assumption that G be 2-connected). Although the proofs of all these results use the same basic idea of gradually transforming one tree into another via a sequence of edge exchanges, some of these processes require sequences that use more properties of trees than do others. We show that the edge exchange proofs can be divided into three types, in accordance with the extent to which the exchange sequence depends upon properties of spanning trees. This idea is then used to obtain new interpolation results for some invariants, and to show how the exchange methods and interpolation results on spanning trees can be extended to other families of spanning subgraphs.
27 citations
14 Jun 1989
TL;DR: A new theory of attribute graph rewriting systems with priorities is developed, which gives an algebraic model which allows us to mathematically prove properties of distributed algorithms.
Abstract: In this paper, we develop a new theory of attribute graph rewriting systems with priorities. This theory provides a very general tool for describing algorithms from classical graph theory, and for algorithms implemented on networks of communicating processors and distributed systems. Moreover, this theory gives an algebraic model which allows us to mathematically prove properties of distributed algorithms.
21 citations
TL;DR: It is shown that if a 2-edge connected graph G has a unique f-factor F, then some vertex has the same degree in F as in G, even if the hypothesis is considerably strengthened.
Abstract: We show that if a 2-edge connected graph G has a unique f-factor F, then some vertex has the same degree in F as in G. This conclusion is the best possible, even if the hypothesis is considerably strengthened.
21 citations
TL;DR: An algorithm for fitting general graphs to proximity data that seeks to provide the connected network that gives the least-squares approximation to the proximity data with the specified number of links, allowing for linear transformations of the data.
Abstract: We present an algorithm for fitting general graphs to proximity data. The algorithm utilizes a mathematical programming procedure based on a penalty function approach to impose additivity constraints upon parameters. For a user-specified number of links, the algorithm seeks to provide the connected network that gives the least-squares approximation to the proximity data with the specified number of links, allowing for linear transformations of the data. The network distance is the minimum-path-length metric for connected graphs. As a limiting case, the algorithm provides a tree where each node corresponds to an object, if the number of links is set equal to the number of objects minus one. A Monte Carlo investigation indicates that the resulting networks tend to fall within one percentage point of the least-squares solution in terms of the variance accounted for, but do not always attain this global optimum. The network model is discussed in relation to ordinal network representations (Klauer 1989) and NETSCAL (Hutchinson 1989), and applied to several well-known data sets.
TL;DR: An O(MIN{m, k2n}n) time algorithm for deciding whether the edge connectivity of a given directed graph G is at least k and both algorithms are superior to the best known algorithms for finding theedge connectivity of directed graphs.
TL;DR: If a practical network is modelled as a graph in which the lines are perfect but the points may fail then a primary measure of vulnerability is the point connectivity of the graph and one possible secondary measure ofulnerability is the number of minimum point disconnecting sets.
Abstract: If a practical network is modelled as a graph in which the lines are perfect but the points may fail then a primary measure of vulnerability is the point connectivity of the graph and one possible secondary measure of vulnerability is the number of minimum point disconnecting sets. The graph shold have the maximum possible point connectivity and the minimum number of point disconnecting sets. A lower bound for the number of minimum point disconnecting sets is derived by identifying points with identical adjacencies.
TL;DR: A connected graph (undirected, all edges of length one) on p vertices such that when a random subset of the vertices are attacked the expected number of vertices that are destroyed (directly and indirectly) is minimized.
Abstract: This paper considers the following variation on the construction of a reliable communication network. Whenever a vertex is attacked, all vertices within distance 2 are also destroyed (or fail) indirectly. We are interested in designing a connected graph (undirected, all edges of length one) on p vertices such that when a random subset of the vertices are attacked the expected number of vertices that are destroyed (directly and indirectly) is minimized. It is assumed that any of the 2p subsets of vertices is equally likely to be attacked. The optimal structure is determined for all p and is shown to be one of five patterns depending on r where p = 5t + r.
TL;DR: For simple graphs F and G, the Ramsey number r(F G) is the smallest integer p such that if the edges of the complete graph KP are colored red and blue either the red subgraph contains a copy of for the blue subgraph containing a copy G as mentioned in this paper.
Abstract: Since it is known that r(K( m ) T) n for the large class of trees that have no vertices of large degree the upper and lower bounds are frequently identical In all cases these bounds are shown to differ by at most k For simple graphs F and G the Ramsey number r(F G) is the smallest integer p such that if the edges of the complete graph KP are colored red and blue either the red subgraph contains a copy of For the blue subgraph contains a copy of G If F is a graph with chromatic number X(F) then the chromatic surplus s(F) is the smallest number of vertices in a color class under any X(F) coloring of the vertices of F For any connected graph G of order n > s(F) the Ramsey number r(F G) satisfies the inequality
TL;DR: For positive integers d, c and v, the minimum number of nodes for a c -connected v -regular graph of diameter d is determined.
TL;DR: Two parallel algorithms for finding maximum bipartite matchings on a CRCW PRAM are presented and simple modifications of these algorithms induce parallel algorithmsfor finding maximum 0–1 flows, which are also presented.
TL;DR: This paper proves the stronger result that G3 - {x,y} is hamiltonian if either x or y is not a cut-vertex of G, and obtains Schaar's characterization of a connected graph G such thatG3 is 2-hamiltonian.
Abstract: This paper deals with the problem of characterizing the pairs of vertices x,y in a connected graph G such that G3 - {x,y} is hamiltonian, where G3 is the cube of G. It is known that the cube G3 is 2-hamiltonian if G is 2-connected. In this paper, we first prove the stronger result that G3 - {x,y} is hamiltonian if either x or y is not a cut-vertex of G, and then proceed to characterize those cut-vertices x and y of G such that G3 -{x,y} is hamiltonian. As a simple consequence of these, we obtain Schaar's characterization of a connected graph G such that G3 is 2-hamiltonian.
01 Dec 1989
TL;DR: It is shown that the graph is super-λ and is an optimal reliable structure for interconnection networks.
Abstract: In this paper, a general class of Boolean n-cube structures is investigated. The interconnection is based on a mixed radix number system which results in a variety of generalized Boolean n-cube structures for a given number of processors N = x·2 y , where x and y are positive integers. A number of interesting properties of the network are revealed. By a constructive method, the node connectivity of this network is found. Finally, we show that the graph is super-λ and is an optimal reliable structure for interconnection networks.
TL;DR: An approach is described in which power system elements can be colored not only with respect to their switch-dependent operating conditions, such as energized or earthed, but also according to the different connectivity units to which they belong.
Abstract: An approach is described in which power system elements can be colored not only with respect to their switch-dependent operating conditions, such as energized or earthed, but also according to the different connectivity units to which they belong. Connectivity units can be nodes, network groups, network districts, and power source groups. Two main issues have to be solved for connectivity unit coloring. First, different colors have to be assigned to neighboring connectivity units. In this way the operator can distinguish at one glance the boundaries of each connectivity unit. Second, modifications caused by switching must be minimized and should reflect the power system changes in a natural way. The task of connectivity unit coloring can be converted to the task of coloring a graph. This enables graph theory to be applied. Algorithms to update graph colors after some vertex-edge changes caused by switch indication changes have never been discussed. Therefore, an innovative update algorithm is provided. Issues in implementing such a power system coloring function within a full graphic-based energy management system are discussed. >
TL;DR: A new metric, called the connectivity function, is presented that allows this kind of consideration to be assessed and can be used to characterize the fault tolerant properties of some common static network topologies.
21 Aug 1989
TL;DR: In this paper, the complexity of connectivity problems on graph languages generated by context-free graph rewriting systems under various restrictions is analyzed, and it is shown that such languages are DEXPTIME-complete w.r.t. log-space reductions.
Abstract: We analyze the precise complexity of connectivity problems on graph languages generated by context-free graph rewriting systems under various restrictions. Let L be the family of all context-free graph rewriting systems that generate at least one disconnected resp. connected graph. We show that L is DEXPTIME-complete w.r.t. log-space reductions. If L is finite then L is PSPACE-complete w.r.t. log-space reductions. These results hold true for graph rewriting systems as for example boundary node label controlled (BNLC) graph grammars, hyper-edge replacement systems (HRS's), apex (APEX) graph grammars, simple context-free node label controlled (SNLC) graph grammars, and even for the simple context-free graph grammars introduced by Slisenko in [Sli82].
TL;DR: In this paper, an upper bound for tree-diameter sets of connected graphs is given, where S (a/sub 1/, a/sub 2/,..., a/ sub n/) is the tree-size set of a connected graph G in increasing order.
Abstract: An upper bound in tree-diameter sets of connected graphs is given. Let S (a/sub 1/, a/sub 2/, . . ., a/sub n/) be the tree-diameter set of a connected graph G in increasing order. It is proved that a/sub i+1/ >
TL;DR: In this article, the authors discuss a related question that arose from the interest in size Ramsey numbers, and show that every finite graph G with minimum degree 6(G) 2 6 contains a spanning bipartite graph H with 6 (H ) 2 la/zl.
Abstract: A classic argument due to Erdos shows that every finite graph G with minimum degree 6(G) 2 6 contains a spanning bipartite graph H with 6 ( H ) 2 la/zl. Jackson [7] has proved that if 6 ( G ) 2 6 z 2, then there exists a batanted spanning bipartite subgraph H with 6 ( H ) 2 1 . Thomassen [ 191, developing the Erdos argument, proved that every finite graph G with 6(G) 2 12k contains a partition ( X , Y ) of V ( C ) such that 6(,Y) 2 k and 6 ( Y ) 2 k . Other papers in this general area include [I] , [3], [S], [ 6 ] , [8]-[14], [20], and [21]. We discuss in this paper an, a t least superficially, related question that arose from our interest [4] in size Ramsey numbers.
01 Apr 1989
TL;DR: In this article, the equivalence classes induced by the transitive closure ΘJ of a connected graph G =(V, E ) were computed in time O(| V | | | E |) and space O(V | 2 ).
Abstract: For two edges e =( x , y ) and e ′=( x ′, y ′) of a connected graph G =( V , E ) let e Θ e ′ iff d ( x , x ′)+ d ( y , y ′)≠ d ( x , y ′)+ d ( x ′, y ). Here d ( x , y ) denotes the lenght of a shortest path in G joining vertices x and y . An algorithm is presented that computes the equivalence classes induced on E by the transitive closure ΘJ of Θ in time O(| V | | E |) and space O(| V | 2 ). Finding the equivalence classes of Θ is the primary step of several graph algorithms.
01 Jan 1989
TL;DR: In this paper, it was shown that the position of the points in the plane is of no interest, neither is the shape of the lines; all that one must avoid is that a line between the points x, y passes through a third point, say z, of the set.
Abstract: Both in mathematics and in its applications, one frequently considers certain pairs of elements of a set. Then it is quite natural to draw the elements of the set as different points of the plane and indicate the considered pairs by lines between the corresponding points. The position of the points in the plane is of no interest, neither is the shape of the lines; all that one must avoid is that a line between the points x, y passes through a third point, say z, of the set. For example, all the drawings in Fig. 1.1 are considered the same; four points are in the set and all pairs but {c, d} occur.
09 Apr 1989
TL;DR: It is concluded that the proposed cluster-seeking algorithm can be used in a computer-vision-related application where the objects of a scene are represented by clusters.
Abstract: A cluster-seeking algorithm incorporating graph-theoretical techniques is proposed. First, a totally connected weighted graph representing the set of patterns under consideration is formed. Second, those edges with weight values greater than a prespecified threshold T are removed from the graph. Next, the disjoint subgraphs corresponding to clusters are identified. Finally, the connectivity strength (or weakness) of each pattern is determined for detecting further separability of clusters. It is concluded that the proposed algorithm can be used in a computer-vision-related application where the objects of a scene are represented by clusters. The patterns can be characterized by the horizontal and vertical position of pixels and the corresponding depth (distance from camera). After applying the algorithm, the clusters are identified as the different objects in the scene. >
TL;DR: The following two theorems are proved: there exists an n -regular n -connected graph G such that for every i ϵ {3,…, g }, G has a cycle of length i if and only if I ϵ I .
01 Jul 1989
TL;DR: A system of rules and techniques is developed for derivation of various classes of parallel algorithms including Systolic algorithms for various fixed connection networks, Randomized parallel algorithms, parallel list ranking, parallel graph connectivity, automatic parallel compilation from segmented straight-line programs, and derivations of pipelined algorithms for small-diameter networks.
Abstract: : A system of rules and techniques is developed for derivation of various classes of parallel algorithms including: 1) Systolic algorithms for various fixed connection networks; 2) Randomized parallel algorithms; 3) Parallel algorithms for tree and graph problems; and 4) Parallel algorithms for algebraic problems. The development is emphasized of fundamental derivation techniques that can be utilized in as wide a class of parallel algorithms as possible. The specific algorithms to be derived have themselves been carefully chosen to be as fundamental as possible. Algorithms and areas currently under investigation include: parallel list ranking, parallel graph connectivity, automatic parallel compilation from segmented straight-line programs, and derivation of pipelined algorithms for small-diameter networks.
01 Jan 1989
TL;DR: Monotonic network analysis (MONA) as mentioned in this paper allows for more general representations of proximity data, and yields a connected graph, weighted by positive integers and possessing a distance function in such a way that the vertices represent the empirical objects, the number of edges is minimal, the weights are minimal, and the ordering of the distances coincides at least approximately (according to some prescribed error criterion) with the order of the dissimilarities.
Abstract: Proximity data can be represented either geometrically or graph theoretically. Graph theoretical methods, however, are typically restricted to representations in terms of trees. As an new method, monotonic network analysis (MONA) allows for more general representations of proximity data. For a given set of data, MONA yields a connected graph, weighted by positive integers and possessing a distance function in such a way that (1) the vertices represent the empirical objects, (2) the number of edges is minimal, (3) the weights are minimal, and (4) the ordering of the distances coincides at least approximately (according to some prescribed error criterion) with the ordering of the dissimilarities. The rationale of MONA will be stated, and the method will be illustrated by applications to real data.