Showing papers on "Path graph published in 1995"
TL;DR: An algorithm, which for fixed k ≥ 0 has running time O (| V(G) | 3 ), to solve the following problem: given a graph G and k pairs of vertices of G, decide if there are k mutually vertex-disjoint paths of G joining the pairs.
1,438 citations
29 May 1995
TL;DR: Polynomial-time approximation algorithms with non-trivial performance guarantees are presented for the problems of partitioning the vertices of a weighted graph into k blocks so as to maximise the weight of crossing edges.
Abstract: Polynomial-time approximation algorithms with non-trivial performance guarantees are presented for the problems of (a) partitioning the vertices of a weighted graph into k blocks so as to maximise the weight of crossing edges, and (b) partitioning the vertices of a weighted graph into two blocks of equal cardinality, again so as to maximise the weight of crossing edges. The approach, pioneered by Goemans and Williamson, is via a semidefinite programming relaxation.
314 citations
TL;DR: The expected time complexity of two graph partitioning problems: the graph coloring and the cut into equal parts is studied to obtain a sublinear expected time algorithm for k-coloring of k-colorable graphs.
209 citations
TL;DR: A polynomial-time approximation algorithm is presented which makes a k -connected graph ( k + 1)-connected by adding a new set of edges with size at most k − 2 over the optimum.
99 citations
TL;DR: A variety of techniques are used to show that the size of the random maximal bipartite graph is quadratic in n but of order only n3'2 in the triangle—free case and a slight improvement in the lower bound of the Ramsey number r(3, t).
Abstract: Let P be a graph property which is preserved by removal of edges. A random maximal P—graph is obtained from n independent vertices by randomly adding edges, at each stage choosing uniformly among edges whose inclusion would not destroy property P, until no further edges can be added. We address the question of the number of edges of a random maximal P—graph for several properties P, in particular the cases of “bipartite” and “triangle—free.” A variety of techniques are used to show that the size of the random maximal bipartite graph is quadratic in n but of order only n3'2 in the triangle—free case. Along the way we obtain a slight improvement in the lower bound of the Ramsey number r(3, t). © 1995 John Wiley & Sons, Inc.
93 citations
TL;DR: A graph-based intermediate representation (IR) with simple semantics and a low-memory-cost C++ implementation that uses inheritance to abstract common opcode behavior, allowing new opcodes to be easily defined from old ones.
Abstract: We present a graph-based intermediate representation (IR) with simple semantics and a low-memory-cost C++ implementation. The IR uses a directed graph with labeled vertices and ordered inputs but unordered outputs. Vertices are labeled with opcodes, edges are unlabeled. We represent the CFG and basic blocks with the same vertex and edge structures. Each opcode is defined by a C++ class that encapsulates opcode-specific data and behavior. We use inheritance to abstract common opcode behavior, allowing new opcodes to be easily defined from old ones. The resulting IR is simple, fast and easy to use.
93 citations
TL;DR: This work first solves the “vertex-disjoint” version of this problem and related questions for edge-colored complete graphs and gives efficient algorithms for finding a fixed number of pairwise vertex- or edge- disjoint paths each of which has given extremities.
39 citations
TL;DR: A sharp lower bound is obtained for the smallest sum of degrees of k + 1 independent vertices of S to be contained in a common cycle of G, which gives a sufficient condition for a k -connected graph to be hamiltonian.
39 citations
TL;DR: It is shown that if G is a connected graph n ≥ 3 vertices such that d(u), d(v) + d(w) n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p (G) - 1, or G is in one of six families of exceptional graphs.
Abstract: For a graph G, p(G) denotes the order of a longest path in G and c(G) the order of a longest cycle. We show that if G is a connected graph n ≥ 3 vertices such that d(u) + d(v) + d(w) n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p(G) - 1, or G is in one of six families of exceptional graphs. This generalizes results of Bondy and of Bauer, Morgana, Schmeichel, and Veldman.
33 citations
TL;DR: A simple linear-time algorithm is presented to compute a dominating path in a connected AT-free graph that avoids the neighborhood of the third.
31 citations
TL;DR: How much does the edge distribution in a triangle-free graph G on n vertices deviate from the uniform edge Distribution in a typical (random) graph on n Vertices with the same number of edges?
01 May 1995
TL;DR: The domination graph of a digraph has the same vertices as the digraph with an edge between two vertices if every other vertex loses to at least one of the two as discussed by the authors.
Abstract: The domination graph of a digraph has the same vertices as the digraph with an edge between two vertices if every other vertex loses to at least one of the two. Previously, the authors showed that the domination graph of a tournament is either an odd cycle with or without isolated and/or pendant vertices, or a forest of caterpillars. They also showed that any graph consisting of an odd cycle with or without isolated and/or pendant vertices is the domination graph of some tournament. This paper extends these results to oriented graphs. We also show that any caterpillar is the domination graph of some digraph, but a path on four or more vertices is not the domination graph of any tournament. Other results relate the domination graph of a tournament to its positive subtournament defined by fisher and Ryan, and the possible and average number of edges in the domination graph of a tournament.
TL;DR: An algorithm to find a c-vertex-ranking of a given tree T using the minimum number of ranks in time O(cn) where n is the number of vertices in T.
TL;DR: It is proved that G is hamiltonian if the degree sum of any κ(G) + 1 pairwise nonadjacent vertices is at least n − λ(G), which is an analog of a result from the thesis of Fournier and generalizes the result of Zhang.
Abstract: Let G be a 2-connected claw-free graph on n vertices, and let H be a subgraph of G. We prove that G has a cycle containing all vertices of H whenever α3(H) ≧ κ(G), where α3(H) denotes the maximum number of vertices of H that are pairwise at distance at least three in G, and κ(G) denotes the connectivity of G. This result is an analog of a result from the thesis of Fournier, and generalizes the result of Zhang that G is hamiltonian if the degree sum of any κ(G) + 1 pairwise nonadjacent vertices is at least n − κ(G). © 1995 John Wiley & Sons, Inc.
TL;DR: Two structural assumptions are established for a plane graph which together guarantee the existence of a triple of pairwise adjacent vertices with restricted degree sum, and it is shown that if any of these assumptions is violated, the degree sum of each three pairwise neighboring vertices may be arbitrarily large.
Patent•
23 May 1995TL;DR: In this paper, a method for operating a FPGA to compute a function whose optimum represents the preferred partitioning of a graph having a plurality of vertices connected by edges is presented.
Abstract: A method for operating a FPGA to compute a function whose optimum represents the preferred partitioning of a graph having a plurality of vertices connected by edges. The FPGA is configured to provide a partition state register having a plurality of cells. Each cell corresponds to one of the vertices in the graph and is used to store a number indicative of the partition to which the corresponding vertex is currently assigned. The algorithm for determining the optimum partition computes a cost function having two components. The assignment of the vertices to the various partitions is made such that this cost function is minimized. For any given assignment of the vertices, the FPGA computes the cost function using two circuits that are configured from the FPGA. The first circuit computes the number of edges that connect vertices belonging to different partitions. The second circuit computes a number that represents the extent to which the various partitions differ from one another in size. The ideal partitioning is that which minimizes a weighted sum of these computed numbers. Special circuits for generating random numbers and binary vectors having a controllable number of randomly placed ones therein are also described.
TL;DR: It is proved that the vertex set V(G) can be decomposed into disjoint subsets V1, ..., Vk so that |Vi| = ni and the subgraph induced by Vi contains no isolated vertices for all i, 1 ≤ i ≤ k.
TL;DR: It was proved that, for every n =f= 1, 3 there exists a cn > 0 such that every minimally n-edge-connected finite simple graph G has at least cn\G\ vertices of degree n.
Abstract: Whereas the number \\G\
of vertices of degree n that a minimally n-connected graph G must have, dependent on the number \\G\\ of vertices of G, is almost exactly known (see [8]), the corresponding problem for minimally n-edge-connected simple graphs is far from being settled. It was shown in [4] that every minimally n-edge-connected finite graph has two vertices of degree n (see also Lemma 13 in [7]), which of course is best possible for every vertex number. But for simple graphs, i.e., graphs without multiple edges, this was improved in [5]: every minimally n-edge-connected finite simple graph has at least n + 1 vertices of degree n. In [6] it was proved that, for every n =f= 1, 3 there exists a cn > 0 such that every minimally n-edge-connected finite simple graph G has at least cn\\G\\ vertices of degree n. For n = 1, 3 such a result does not hold, as the example in Figure 1 shows for n = 3. The value of the constant cn was improved in [1] and [2], and a rather good estimate for \\G\
was given quite recently by Cai Mao-Cheng [3].
TL;DR: It is shown that if in a simple graph G of order n the sum of degrees of any three pairwise non-adjacent vertices is at least n, then there are two cycles of GF that contain all the vertices.
Abstract: It is shown that if in a simple graph G of order n the sum of degrees of any three pairwise non-adjacent vertices is at least n, then there are two cycles (or one cycle and an edge or a vertex) of GF that contain all the vertices. © 1995 John Wiley & Sons, Inc.
TL;DR: It is shown that there is a cycle in G that contains all vertices of degree k and that it is possible to write a graph on n vertices with maximum degree k where n ≤ 3k - 2.
Abstract: Let G be a 2-connected graph on n vertices with maximum degree k where n ≤ 3k - 2. We show that there is a cycle in G that contains all vertices of degree k. © 1995 John Wiley & Sons, Inc.
TL;DR: The minimum of the largest weights assigned over all such irregular assignments on the union of t copies of the complete graph with p vertices, p ⩾ 3, is determined.
TL;DR: In this article, the edge number of a graph of order n ≥ 517 ensures that it contains every complete graph and every forest with at most n vertices and at most m edges.
TL;DR: It is shown that every 2-connected, k-regular claw-free graph on n vertices contains a cycle of length at least min 4k?2, n (k?8), and this result is best possible.
04 Jan 1995
TL;DR: Experimental results on several large benchmarks and production VLSI circuits show that the new, fast algorithm for optimally retiming large sequential circuits under the unit delay model is significantly faster than the best optimal Retiming method known to date.
Abstract: We present a new, fast algorithm for optimally retiming large sequential circuits under the unit delay model. Our method consists of two main steps: (1) computation of the optimum clock period and (2) computation of a feasible retiming For the optimum clock period. We construct a path graph that has as many vertices as there are flip-flops in the circuit. The path graph also has an additional vertex that corresponds to all primary Inputs and outputs of the circuit. There is an arc from a vertex to another if there is a strictly combinational path between the corresponding flip-flops, primary inputs or outputs. We formulate an integer linear program (ILP) on the path graph to compute the minimum clock period /spl phi//sub opt/ for which the path graph has no critical cycles. An optimum solution to the ILP is determined from the optimum solution of the corresponding linear program (LP) relaxation. We show that /spl phi//sub opt/ is also the optimum clock period for the circuit. After determining the optimum clock period, a feasible retiming for the optimum clock period is obtained using known retiming methods. Experimental results on several large benchmarks and production VLSI circuits show that our method is significantly faster than the best optimal retiming method known to date. Also, optimum retiming results for these benchmark circuits are being presented for the first time.
TL;DR: For all oddv ≥ 3 the complete graph onvKv vertices can be decomposed intov − 2 edge disjoint cycles whose lengths are 3, 3, 4, 5,...,v − 1.
Abstract: For all odd upsilon greater than or equal to 3 the complete graph on upsilon K(u)psilon vertices can be decomposed into upsilon - 2 edge disjoint cycles whose lengths are 3, 3, 4, 5,..., upsilon - 1. Also, for all odd upsilon greater than or equal to 7, K(u)psilon can be decomposed into upsilon - 3 edge disjoint cycles whose lengths are 3, 4,..., upsilon - 4, upsilon - 2, upsilon - 1, upsilon.
TL;DR: In this article, it was shown that the dimension of the class of directed rectangle-visibility graphs is unbounded, which is the only known dimension of directed acyclic graphs.
TL;DR: A unified argument is given to show that asymptotically, almost all full graphs can be obtained by taking an arbitrary undirected graph in the n vertices, distinguishing a clique in this graph that need not be maximal, and then adding directed edges going out from each vertex in the clique to all vertices to which there is not already an existing undirecting edge.
Abstract: A full graph on n vertices, as defined by Fulkerson, is a representation of the intersection and containment relations among a system of n sets. It has an undirected edge between vertices representing intersecting sets, and a directed edge from a to b if the corresponding set A contaisn B. We give a unified argument to show that asymptotically, almost all full graphs can be obtained by taking an arbitrary undirected graph in the n vertices, distinguishing a clique in this graph that need not be maximal, and then adding directed edges going out from each vertex in the clique to all vertices to which there is not already an existing undirected edge. © 1996 John Wiley & Sons, Inc.
TL;DR: This paper investigates the conjecture that a graph is perfect if it admits a two-edge-coloring such that two edges receive different colors if they are the nonincident edges of a P4 (chordless path with four vertices).
Abstract: We investigate the conjecture that a graph is perfect if it admits a two-edge-coloring such that two edges receive different colors if they are the nonincident edges of a P4 (chordless path with four vertices). Partial results on this conjecture are given in this paper. © 1995 John Wiley & Sons, Inc.
TL;DR: It is shown that every 1-tough graph G on n ≥ 3 vertices with σ3≧ n has a cycle of length at least min{n, n + (σ3/3 ) − α + 1}, where �o3 denotes the minimum value of the degree sum of any 3 pairwise nonadjacent vertices and α the cardinality of a miximum independent set of vertices in G.
Abstract: We show that every 1-tough graph G on n ≥ 3 vertices with σ3≧ n has a cycle of length at least min{n, n + (σ3/3 ) − α + 1}, where σ3 denotes the minimum value of the degree sum of any 3 pairwise nonadjacent vertices and α the cardinality of a miximum independent set of vertices in G. Our inequality is sharp and implies some sufficient conditions of hamiltonian cycles. © 1995 John Wiley & Sons, Inc.
TL;DR: It is proved that, in a distance-regular interval-regular graph having a triangle, I(u,v) is convex for every vertices u,v at distance 3.