Showing papers on "Path graph published in 2001"
TL;DR: Here the authors obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3.9±0.1 is obtained.
Abstract: Recently, Barabasi and Albert [2] suggested modeling complex real-world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabasi and Albert suggested that after many steps the proportion P(d) of vertices with degree d should obey a power law P(d)αd−γ. They obtained γ=2.9±0.1 by experiment and gave a simple heuristic argument suggesting that γ=3. Here we obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 279–290, 2001
891 citations
TL;DR: It is concluded that grown graphs, however randomly they are constructed, are fundamentally different from their static random graph counterparts.
Abstract: We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability $\ensuremath{\delta},$ two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time steps. In the limit of large t, the resulting graph displays surprisingly rich characteristics. In particular, a giant component emerges in an infinite-order phase transition at $\ensuremath{\delta}=1/8.$ At the transition, the average component size jumps discontinuously but remains finite. In contrast, a static random graph with the same degree distribution exhibits a second-order phase transition at $\ensuremath{\delta}=1/4,$ and the average component size diverges there. These dramatic differences between grown and static random graphs stem from a positive correlation between the degrees of connected vertices in the grown graph---older vertices tend to have higher degree, and to link with other high-degree vertices, merely by virtue of their age. We conclude that grown graphs, however randomly they are constructed, are fundamentally different from their static random graph counterparts.
414 citations
TL;DR: It is shown that if a graph has a vertex cover of size k, then one can select in polynomial time a set of k vertices that covers over 80% of the edges.
181 citations
TL;DR: It is shown that the size of the largest component of the graph formed at stage 0.535n is polylogarithmic in n, which resolves a question of Achlioptas.
Abstract: Let e1, e′1; e2, e′2;…;ei, e′i;⋅⋅⋅ be a sequence of ordered pairs of edges chosen uniformly at random from the edge set of the complete graph Kn (i.e. we sample with replacement). This sequence is used to form a graph by choosing at stage i, i=1,…, one edge from ei,e′i to be an edge in the graph, where the choice at stage i is based only on the observation of the edges that have appeared by stage i. We show that these choices can be made so that whp the size of the largest component of the graph formed at stage 0.535n is polylogarithmic in n. This resolves a question of Achlioptas. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 75–85, 2001
131 citations
TL;DR: In this paper, the authors studied both numerically and analytically what happens to a random graph of average connectivity α when its leaves and their neighbors are removed iteratively up to the point when no leaf remains.
Abstract: We study both numerically and analytically what happens to a random graph of average connectivity α when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the core. In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated vertices and the number of vertices and edges in the core. We show that a second order phase transition occurs at α = e = 2.718 ... : below the transition, the core is small but above the transition, it occupies a finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in the critical region, and we propose a consistent set of critical exponents, which does not coincide with the set of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization and spectral properties of the adjacency matrix of random graphs.
94 citations
TL;DR: Three algorithms for finding small-stretch paths between all pairs of vertices in a weighted graph with n vertices and m edges are described.
79 citations
TL;DR: This paper aims to embed longest fault-free paths in an n-dimensional star graph with edge faults, and discusses the situation of n<6, where n-3 (edge faults) is maximal in the worst case.
Abstract: In this paper, we aim to embed longest fault-free paths in an n-dimensional star graph with edge faults. When n/spl ges/6 and there are n-3 edge faults, a longest fault-free path can be embedded between two arbitrary distinct vertices, exclusive of two exceptions in which at most two vertices are excluded. Since the star graph is regular of degree n-1, n-3 (edge faults) is maximal in the worst case. When n/spl ges/6 and there are n-4 edge faults, a longest fault-free path can be embedded between two arbitrary distinct vertices. The situation of n<6 is also discussed.
60 citations
TL;DR: It is shown that all graphs in some classes, including all trees, are stronglymultiplicative, and the question of the maximum number of edges in a strongly multiplicative graph of a given order is considered.
Abstract: A graph with p vertices is said to be strongly multiplicative if its vertices can be labelled 1;2;:::;p so that the values on the edges, obtained as the product of the labels of their end vertices, are all distinct In this paper, we study structural properties of strongly multiplicative graphs We show that all graphs in some classes, including all trees, are strongly multiplicative, and consider the question of the maximum number of edges in a strongly multiplicative graph of a given order
60 citations
28 Aug 2001
TL;DR: It is proved that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth, and a path decomposition is used as the basis for a linear-time algorithm to decide if a graph has a crossing- free h- layer drawing.
Abstract: We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.
59 citations
TL;DR: The inequality α(G)⩾ 1 2 [(2m+n+1)− (2m-n-1) 2 −4n 2 ] is proved and its algorithmic realization is discussed.
53 citations
TL;DR: In this paper, the maximum size of a subset of vertices that induces a forest is defined as the maximum number of edges that can be connected with n vertices, e edges, and maximum degree Δ.
Abstract: For a graph G, let a(G) denote the maximum size of a subset of vertices that induces a forest. Suppose that G is connected with n vertices, e edges, and maximum degree Δ. Our results include: (a) if Δ ≤ 3, and G ≠ K4, then a(G) ≥ n - e-4 - 1-4 and this is sharp for all permissible e ≡ 3 (mod 4); and (b) if Δ ≥ 3, then a(G) ≥ α(G) + (n - α(G))-(Δ - 1)2. Several problems remain open. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 113–123, 2001
TL;DR: The average distance μ of a connected graph with n vertices, e edges and minimum degree d satisfies μ⩽ ⌊ (n+1)n(n−1)−2e d+1 ⌋ n(n −1) .
TL;DR: In this article, it was shown that between any two vertices of a 5-connected graph there exists an induced path whose vertices can be removed such that the remaining graph is 2-connected.
Abstract: We show that between any two vertices of a 5-connected graph there exists an induced path whose vertices can be removed such that the remaining graph is 2-connected. © 2000 John Wiley & Sons, Inc. J Graph Theory 36: 52–58, 2001
13 Jun 2001
TL;DR: Improved semidefinite programming based approximation algorithms are obtained for all the natural maximum bisection problems of graphs and improve, extend and unify results of Frieze and Jerrum, Feige and Langberg, Ye, and others.
Abstract: We obtain improved semidefinite programming based approximation algorithms for all the natural maximum bisection problems of graphs. Among the problems considered are: MAX n/2 -BISECTION - partition the vertices of a graph into two sets of equal size such that the total weight of edges connecting vertices from different sides is maximized; MAX n/2 -VERTEX-COVER - find a set containing half of the vertices such that the total weight of edges touching this set is maximized; MAX n/2 -DENSE-SUBGRAPH - find a set containing half of the vertices such that the total weight of edges connecting two vertices from this set is maximized; and MAX n/2 -UnCUT - partition the vertices into two sets of equal size such that the total weight of edges that do not cross the cut is maximized. We also consider the directed versions of these problems, MAX n/2 -DIRECTED-BISECTION and MAX n/2 -DIRECTED-UnCUT. These results can be used to obtain improved approximation algorithms for the unbalanced versions of the partition problems mentioned above, where we want to partition the graph into two sets of size k and n - k, where k is not necessarily n/2 . Our results improve, extend and unify results of Frieze and Jerrum, Feige and Langberg, Ye, and others.
TL;DR: To overcome the difficulty arising from random selection of the two end vertices, a new embedding method, based on a backtracking technique, is proposed and can tolerate more edge faults than Hsieh et al.
Abstract: The arrangement graph, denoted by A/sub n,k/, is a generalization of the star graph. A recent work by S.Y. Hsieh et al. (1999) showed that when n-k/spl ges/4 and k=2 or n-k/spl ges/4+[k/2] and k/spl ges/3, A/sub n,k/ with k(n-k)-2 random edge faults, can embed a Hamiltonian cycle. In this paper, we generalize Hsieh et al. work by embedding a Hamiltonian path between arbitrary two distinct vertices of the same A/sub n,k/. To overcome the difficulty arising from random selection of the two end vertices, a new embedding method, based on a backtracking technique, is proposed. Our results can tolerate more edge faults than Hsieh et al. results as k/spl ges/7 and 7/spl les/n-k/spl les/3+[k/2], although embedding a Hamiltonian path between arbitrary two distinct vertices is more difficult than embedding a Hamiltonian cycle.
TL;DR: It is proved that a noncomplete ?-connected graph for which the sum of the degrees of any two distinct vertices is at least 2?54???1 possesses a ?-contractible edge.
TL;DR: It is shown that triangle-free outerplanar graphs instantiate the results and speculate on other families of graphs that might have r-color extension theorems.
TL;DR: Borders on α ( G) are given for various families of sparse graphs, including trees, planar graphs and graphs with bounded arboricity.
TL;DR: A broadcast graph is constructed on 26 vertices with 42 edges to prove B(26) = 42 to prove the minimum number of edges in any broadcast graph on n vertices.
TL;DR: The chromatic spectrum of G is the set of all vertices of G for which G has a strict $k$-coloring and it is shown that a set of integers S is the spectrum of some plane graph with face-constraints if and only if $S$ is an interval with a gap.
Abstract: Let $G$ be a plane graph whose vertices are to be colored subject to constraints on some of the faces. There are 3 types of constraints: a $C$ indicates that the face must contain two vertices of a $C$ommon color, a $D$ that it must contain two vertices of a $D$ifferent color and a $B$ that $B$oth conditions must hold simultaneously. A coloring of the vertices of $G$ satisfying the facial constraints is a strict $k$-coloring if it uses exactly $k$ colors. The chromatic spectrum of $G$ is the set of all $k$ for which $G$ has a strict $k$-coloring. We show that a set of integers $S$ is the spectrum of some plane graph with face-constraints if and only if $S$ is an interval $\{s,s+1,\dots,t\}$ with $1\leq s\leq 4$, or $S=\{2,4,5,\dots,t\}$, i.e. there is a gap at 3.
TL;DR: It is proved that every 1-tough graph has a partition of its vertices into paths of length at least two.
TL;DR: It is shown that, for any vertex subset A with at most n/2d vertices, there exists a Hamiltonian cycle in which the distance between any two vertices of A is at least d.
TL;DR: O(n3) algorithms to recognize HH- free graphs and HHD-free graphs are presented and the previous best algorithms for the problems run in O(n4) time.
23 Sep 2001
TL;DR: It is shown that there exists a recursive function f( n) with the property that every string graph of n vertices has a representation in which any two curves cross at most f(n) times.
Abstract: A graph is called a string graph if its vertices can be represented by continuous curves (“strings”)in the plane so that two of them cross each other if and only if the corresponding vertices are adjacent. It is shown that there exists a recursive function f(n)with the property that every string graph of n vertices has a representation in which any two curves cross at most f(n)times. We obtain as a corollary that there is an algorithm for deciding whether a given graph is a string graph. This solves an old problem of (1959), (1966), and G(1971).
TL;DR: In this paper, it was shown that if G is a 3-connected graph of order n such that the circumference of G satisfies c(G) ≥ p(G)-1, then G satisfies ρ ≥ ρ - 1 for every independent set {x1, x2, x3, x4}.
Abstract: For a graph G, let p(G) denote the order of a longest path in G and c(G) the order of a longest cycle in G, respectively. We show that if G is a 3-connected graph of order n such that $\textstyle{\sum^{4}_{i=1}\,{\rm deg}_{G}\,x_{i} \ge {3\over2}\,n + 1}$ for every independent set {x1, x2, x3, x4}, then G satisfies c(G) ≥ p(G) - 1. Using this result, we give several lower bounds to the circumference of a 3-connected graph. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 137156, 2001
TL;DR: In this article, the authors generalize Hsieh et al.'s result by constructing a longest fault-free path between two arbitrary distinct healthy vertices, which is a generalization of the star graph.
Abstract: The arrangement graph is a generalization of the star graph. Recent work by Hsieh et al. showed that an arrangement graph with faulty vertices contains a longest fault-free cycle. In this paper, we generalize Hsieh et al.'s result by constructing a longest fault-free path between two arbitrary distinct healthy vertices. Moreover, our result can tolerate more faulty vertices than can Hsieh et al.'s result. The situation of simultaneous faulty vertices and faulty edges is also discussed.
TL;DR: It is proved that each 3-connected plane graph G without triangular or quadrangular faces either contains a k-path Pk, a path on k vertices, such that each of its k vertice has degree ≤5/3k in G or does not contain any k- path.
Abstract: We prove that each 3-connected plane graph G without triangular or quadrangular faces either contains a k-path Pk, a path on k vertices, such that each of its k vertices has degree ≤5/3k in G or does not contain any k-path. We also prove that each 3-connected pentagonal plane graph G which has a k-cycle, a cycle on k vertices, k∈ {5,8,11,14}, contains a k-cycle such that all its vertices have, in G, bounded degrees. Moreover, for all integers k and m, k≥ 3, k∉ {5,8,11,14} and m≥ 3, we present a graph in which every k-cycle contains a vertex of degree at least m.
TL;DR: It is proved that, over all graphs G =( V; E) with 5xed |V |?3 and |E|, the average of � (G) is at least |E |− 3|V |(log|V|)=[log(( |V| 2 )=|E|)] −| V |− 1.
TL;DR: A review on the progress made in the last 25 years on the Reconstruction Conjecture, and discussing the class of all finite trees, goes back to the original Kelly's interest.
TL;DR: This paper studies the behavior of diam as a function of i, where P 2 i (G) is a composition P 2 (P 2 i−1 (G)) , with P 2 0 (G)=G .