Showing papers on "Path graph published in 2000"
TL;DR: A polynomial time approximation algorithm is presented for problems modeled by the divide-and-conquer paradigm, such that all subgraphs for which the optimization problem is nontrivial have large diameters.
Abstract: We present a novel divide-and-conquer paradigm for approximating NP-hard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divide-and-conquer approach is applicable. Second, a fractional spreading metric is computable in polynomial time. The spreading metric assigns lengths to either edges or vertices of the input graph, such that all subgraphs for which the optimization problem is nontrivial have large diameters. In addition, the spreading metric provides a lower bound, τ, on the cost of solving the optimization problem. We present a polynomial time approximation algorithm for problems modeled by our paradigm whose approximation factor is O(min{log τ, log log τ, log k log log k}) where k denotes the number of “interesting” vertices in the problem instance, and is at most the number of vertices.We present seven problems that can be formulated to fit the paradigm. For all these problems our algorithm improves previous results. The problems are: (1) linear arrangement; (2) embedding a graph in a d-dimensional mesh; (3) interval graph completion; (4) minimizing storage-time product; (5) subset feedback sets in directed graphs and multicuts in circular networks; (6) symmetric multicuts in directed networks; (7) balanced partitions and p-separators (for small values of p) in directed graphs.
171 citations
TL;DR: Two algorithms for listing all simplicial vertices of a graph running in time O (n α ) and O (e 2α/(α+1) )= O ( e 1.41) are given, and it is shown that counting the number of K 4 's in a graph can be done in timeO (e ( α+1)/2 ) .
124 citations
TL;DR: In this paper, it was shown that the star graphs with dimension four or larger are strongly Hamiltonian-laceable, i.e., they have a longest path between every two of their vertices.
Abstract: Suppose that G is a bipartite graph with its partite sets of equal size. G is said to be strongly Hamiltonian-laceable if there is a Hamiltonian path between every two vertices that belong to different partite sets and there is a path of (maximal) length N - 2 between every two vertices that belong to the same partite set, where N is the order of G. In other words, a strongly Hamiltonian-laceable graph has a longest path between every two of its vertices. In this paper, we show that the star graphs with dimension four or larger are strongly Hamiltonian-laceable. © 2000 john Wiley & Sons, Inc.
102 citations
TL;DR: This paper gives a set of 57 graphs and proves that it is the set of the minimal forbidden minors for the class of graphs with linear-width at most two and gives a linear time algorithm that either reports that a given graph has linear- width more than two or outputs an edge ordering of minimumlinear-width.
69 citations
TL;DR: A family of simple incremental algorithms for constructing short paths on the Delaunay graph is defined and potential applications to routeing in mobile communication networks are discussed.
Abstract: Consider the Delaunay graph and the Voronoi tessellation constructed with respect to a Poisson point process. The sequence of nuclei of the Voronoi cells that are crossed by a line defines a path on the Delaunay graph. We show that the evolution of this path is governed by a Markov chain. We study the ergodic properties of the chain and find its stationary distribution. As a corollary, we obtain the ratio of the mean path length to the Euclidean distance between the end points, and hence a bound for the mean asymptotic length of the shortest path. We apply these results to define a family of simple incremental algorithms for constructing short paths on the Delaunay graph and discuss potential applications to routeing in mobile communication networks.
67 citations
TL;DR: This paper gives embeddings of complete binary trees into square grids and extended grids with total vertex-congestion 1, i.e., for any vertex x of the extended grid the authors have load(x)+vertex-Congestion(x)⩽1.
59 citations
18 Dec 2000
TL;DR: A constructive proof of the fact that both the Graph Minor series of Robertson and Seymour imply and the algorithms of this proof are optimal and able to output the corresponding pair (T, χ) in case of an affirmative answer are given.
Abstract: Consider the following problem: For any constant k and any input graph G, check whether there exists a tree T with internal vertices of degree 3 and a bijection χ mapping the vertices of G to the leaves of T such that for any edge of T, the number of edges of G whose end-points have preimages in different components of T - e, is bounded by k. This problem is known as the MINIMUM ROUTING TREE CONGESTION problem and is relevant to the design of minimum congestion telephone networks. If, in the above definition, we consider lines instead of trees with internal vertices of degree 3 and bijections mapping the vertices of G to all the vertices of T, we have the well known MINIMUM CUT LINEAR ARRANGEMENT problem. Recent results of the Graph Minor series of Robertson and Seymour imply (non-constructively) that both these problems are fixed parameter tractable. In this paper we give a constructive proof of this fact. Moreover, the algorithms of our proof are optimal and able to output the corresponding pair (T, χ) in case of an affirmative answer.
53 citations
TL;DR: In this article, it was shown that max d i +d j −|N i ∩N j |:1⩽i is an upper bound for the largest eigenvalue of the Laplacian matrix of a graph.
49 citations
Patent•
18 Aug 2000TL;DR: In this paper, a method and system for designing a network includes generating a representation of a candidate network, where each vertex represents a path and each edge couples at least two vertices representing paths of which at most one path can be included in a network.
Abstract: A method and system for designing a network includes generating a representation of a candidate network. The representation includes vertices and edges, where each vertex represents a path and each edge couples at least two vertices representing paths of which at most one path can be included in a network. A set of a maximum number of vertices, where no two vertices are coupled by an edge, is determined. The paths represented by the vertices of the set are included in the network.
42 citations
TL;DR: It is shown that a geometric graph of n vertices with no k+1 pairwise disjoint edges has at most 29k2n edges.
41 citations
TL;DR: It is shown that polynomial solvability of the maximum stable set problem in P 5 -free banner-free graphs, where P 5 is the simple path on five vertices and a banner is the graph with vertices a, b, c, d, e and edges.
TL;DR: This paper presents several approximation algorithms for the shortest total path length spanning tree problem and shows that the approximation ratio of (4=3+) can be achieved in polynomial time for any constant >0.
TL;DR: Aldous et al. as discussed by the authors studied the emergence of giant components with O(n 2/3) vertices in the Erdos-Renyi graph process with the number of vertices fixed at n at the start.
Abstract: Author(s): Aldous, DJ; Pittel, B | Abstract: A randomly evolving graph, with vertices immigrating at rate n and each possible edge appearing at rate 1/n, is studied. The detailed picture of emergence of giant components with O(n2/3) vertices is shown to be the same as in the Erdos-Renyi graph process with the number of vertices fixed at n at the start. A major difference is that now the transition occurs about a time t = π/2, rather than t = 1. The proof has three ingredients. The size of the largest component in the subcritical phase is bounded by comparison with a certain multitype branching process. With this bound at hand, the growth of the sum-of-squares and sum-of-cubes of component sizes is shown, via martingale methods, to follow closely a solution of the Smoluchowsky-type equations. The approximation allows us to apply results of Aldous [Brownian excursions, critical random graphs and the multiplicative coalescent, Ann Probab 25 (1997), 812-854] on emergence of giant components in the multiplicative coalescent, i.e., a nonuniform random graph process. © 2000 John Wiley a Sons, Inc. Random Struct. Alg., 17, 79-102, 2000.
TL;DR: Progress on closures made in the past (more than) twenty years is surveyed, finding that many authors developed other closure concepts for a variety of graph properties, or used closure techniques as a tool for obtaining deeper sufficiency results with respect to these properties.
Abstract: In this paper we survey results of the following type (known as closure results). Let P be a graph property, and let C(u,v) be a condition on two nonadjacent vertices u and v of a graph G. Then G+uv has property P if and only if G has property P. The first and now well-known result of this type was established by Bondy and Chvatal in a paper published in 1976: If u and v are two nonadjacent vertices with degree sum n in a graph G on n vertices, then G+uv is hamiltonian if and only if G is hamiltonian. Based on this result, they defined the n-closure cln (G) of a graph G on n vertices as the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree sum n until no such pair remains. They showed that cln(G) is well-defined, and that G is hamiltonian if and only if cln(G) is hamiltonian. Moreover, they showed that cln(G) can be obtained by a polynomial algorithm, and that a Hamilton cycle in cln(G) can be transformed into a Hamilton cycle of G by a polynomial algorithm. As a consequence, for any graph G with cln(G)=Kn (and n≥3), a Hamilton cycle can be found in polynomial time, whereas this problem is NP-hard for general graphs. All classic sufficient degree conditions for hamiltonicity imply a complete n-closure, so the closure result yields a common generalization as well as an easy proof for these conditions. In their first paper on closures, Bondy and Chvatal gave similar closure results based on degree sum conditions for nonadjacent vertices for other graph properties. Inspired by their first results, many authors developed other closure concepts for a variety of graph properties, or used closure techniques as a tool for obtaining deeper sufficiency results with respect to these properties. Our aim is to survey this progress on closures made in the past (more than) twenty years.
TL;DR: It is proved the sharper result that p(G)⩽ 1 2 u+⌊ 2 3 g⌋ where u is the number of odd vertices and g is theNumber of nonisolated even vertices.
TL;DR: It is proved that for sufficiently large k, every k-critical triangle-free graph on n vertices has at least (k-o(k))n edges, and it is shown that every (k+1)-critical hypergraph onn vertices and without graph edges has at at least edges.
Abstract: is called k-critical if it has chromatic number k, but every proper sub(hyper)graph of it is (k-1)-colourable. We prove that for sufficiently large k, every k-critical triangle-free graph on n vertices has at least (k-o(k))n edges. Furthermore, we show that every (k+1)-critical hypergraph on n vertices and without graph edges has at least edges. Both bounds differ from the best possible bounds by o(kn) even for graphs or hypergraphs of arbitrary girth.
TL;DR: It is proved that every 3-connected planar graph G with δ ( G )⩾4 either does not contain any path on k ⩾8 vertices or must contain a path onk vertices having degree 5 k −7; the bound 5k −7 is shown to be the best possible.
TL;DR: It will be proved that G contains n vertex-disjoint triangles, and it will be shown by example that this is close to being sharp.
TL;DR: Taking these weights as independent normal N(p, pq) random variables gives a ‘continuous’ approximation to [Gscr ](n, p) whose degrees are much easier to analyse.
Abstract: Let 0 < p < 1, q = 1 − p and b be fixed and let G ∈ G(n, p) be a graph on n vertices where each pair of vertices is joined independently with probability p. We show that the probability that every vertex of G has degree at most pn + b √npq is equal to (c + o(1))n, for c = c(b) the root of a certain equation. Surprisingly, c(0) = 0.6102 … is greater than ½ and c(b) is independent of p. To obtain these results we consider the complete graph on n vertices with weights on the edges. Taking these weights as independent normal N(p, pq) random variables gives a ‘continuous’ approximation to G(n, p) whose degrees are much easier to analyse.
TL;DR: This work presents sequential and parallel algorithms to find an optimal c -vertex-ranking of a partial k -tree, that is, a graph of treewidth bounded by a fixed integer k .
TL;DR: It is proved that G has an [ a, b ]-factor if the minimum degree δ ( G )⩾ a, n ⩾2( a + b )( a+ b −1)/ b and | N G ( x )∪N G ( y )|⩽ an /( a - b ) for any two non-adjacent vertices x and y of G.
TL;DR: In this paper, an optimal consistent fault-tolerant routing of shortest paths is defined for a chordal ring graph with odd diameter and maximum order, which is accomplished by associating to the graph a triple-loop one.
Abstract: This paper studies routing vulnerability in networks modeled by chordal ring graphs. In a chordal ring graph, the vertices are labeled in ℤ2n and each even vertex i is adjacent to the vertices i + a, i + b, i, + c, where a, b, and c are different odd integers. Our study is based on a geometrical representation that associates to the graph a tile which periodically tessellates the plane. Using this approach, we present some previous results on triple-loop graphs, including an algorithm to calculate the coordinates of a given vertex in the tile. Then, an optimal consistent fault-tolerant routing of shortest paths is defined for a chordal ring graph with odd diameter and maximum order. This is accomplished by associating to the chordal ring graph a triple-loop one. When some faulty elements are present in the network, we give a method to obtain central vertices, which are vertices that can be used to reroute any communication affected by the faulty elements. This implies that the diameter of the corresponding surviving route graph is optimum. © 2000 John Wiley & Sons, Inc.
TL;DR: In this article, it was shown that if G is a 3-connected graph or order n and if S is a subset of vertices such that the degree sum of any four independent vertices of S is at least n 1 + 2 α(S, G) − 2, then S is cyclable.
Abstract: A subset S of vertices of a graph G is called cyclable in G if there is in G some cycle containing all the vertices of S. We denote by α(S, G) the number of vertices of a maximum independent set of G[S]. We prove that if G is a 3-connected graph or order n and if S is a subset of vertices such that the degree sum of any four independent vertices of S is at least n + 2α(S, G) -2, then S is cyclable. This result implies several known results on cyclability or Hamiltonicity. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 191–203, 2000
The work was done while this author was visiting L.R.I. by a cooperation program between CNRS and Academia Sinica and was partially supported by NNSF of China.
TL;DR: This paper gives a good characterization for the existence of a simple graph G with vertices v 1 ,v 2 ,…,v n such that a i ⩽d G (v i )⩽b i for i=1,2,…,n .
TL;DR: In this article, the minimum degree sum of a pair of nonadjacent vertices in a graph G was conjectured and verified for the cases where almost all ai ≥ 5 and k ≥ 3.
Abstract: For a graph G, let σ2(G) denote the minimum degree sum of a pair of nonadjacent vertices. We conjecture that if |V(G)| = n = Σki = 1 ai and σ2(G) ≥ n + k - 1, then for any k vertices v1, v2,…, vk in G, there exist vertex-disjoint paths P1, P2,…, Pk such that |V(Pi)| = ai and vi is an endvertex of Pi for 1 ≤ i ≤ k. In this paper, we verify the conjecture for the cases where almost all ai ≤ 5, and the cases where k ≤ 3. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 163–169, 2000
TL;DR: In this article, the average case complexity of shortest paths in the vertex-potential model was studied and it was shown that on a graph with n vertices and with respect to this model, the single-source shortest-paths problem can be solved in O(n 2 ) expected time.
Abstract: We study the average-case complexity of shortest paths problems in the vertex-potential model. The vertex-potential model is a family of probability distributions on complete directed graphs with arbitrary real edge lengths but without negative cycles. We show that on a graph with n vertices and with respect to this model, the single-source shortest-paths problem can be solved in O(n2) expected time, and the all-pairs shortest-paths problem can be solved in O(n2 log n) expected time.
TL;DR: It is proved that the greedy (First-Fit) on-line algorithm, assigning the smallest feasible color to the next vertex at each step, generates a (3 log 2 n) -ranking for the path with n ⩾2 vertices, independently of the order in which the vertices are received.
TL;DR: It is proved that, for r? 2 andn?n(r), every directed graph with n vertices and more edges than the r -partite Turan graph T(r, n) contains a subdivision of the transitive tournament on r+ 1 vertices.
Abstract: We prove that, for r? 2 andn?n(r), every directed graph with n vertices and more edges than the r -partite Turan graph T(r, n) contains a subdivision of the transitive tournament on r+ 1 vertices. Furthermore, the extremal graphs are the orientations ofT (r, n) induced by orderings of the vertex classes.
TL;DR: In this paper, the authors considered the family of trees with n vertices and a fixed maximum vertex degree, and derived some methods that can strictly reduce the Wiener number by shifting leaves.
Abstract: The Wiener number (W) of a connected graph is the sum of distances for all pairs of vertices. As a graphical invariant, it has been found extensive application in chemistry. Considering the family of trees with n vertices and a fixed maximum vertex degree, we derive some methods that can strictly reduceW by shifting leaves. And then, by a process, we prove that the dendrimer on n vertices is the unique graph reaching the minimum Wiener number. c 2000 John Wiley & Sons, Inc. Int J Quantum Chem 78: 331-340, 2000
TL;DR: It is proved that the graph of bistellar flips between triangulations of a vector configuration A with d+4 elements in rank d+1 is connected in general and 3-connected for acyclic vector configurations, which include all point configurations of dimension d with d-4 elements.
Abstract: We study the graph of bistellar flips between triangulations of a vector configuration A with d+4 elements in rank d+1 (i.e. with corank 3), as a step in the Baues problem. We prove that the graph is connected in general and 3-connected for acyclic vector configurations, which include all point configurations of dimension d with d+4 elements. Hence, every pair of triangulations can be joined by a finite sequence of bistellar flips and, in the acyclic case, every triangulation has at least three geometric bistellar neighbours. In corank 4, connectivity is not known and having at least four flips is false. In corank 2, the results are trivial since the graph is a cycle. Our methods are based on a dualization of the concept of triangulation of a point or vector configuration A to that of a virtual chamber of its Gale transform B , introduced by de Loera et al. in 1996. As an additional result we prove a topological representation theorem for virtual chambers, stating that every virtual chamber of a rank 3 vector configuration B can be realized as a cell in some pseudo-chamber complex of B in the same way that regular triangulations appear as cells in the usual chamber complex. All the results in this paper generalize to triangulations of corank 3 oriented matroids and virtual chambers of rank 3 oriented matroids, realizable or not. The details for this generalization are given in the Appendix.