Showing papers on "Path graph published in 2004"

The Diameter of a Scale-Free Random Graph

Abstract: We consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree. This process was introduced by Barabasi and Albert [3], as a simple model of the growth of real-world graphs such as the world-wide web. Computer experiments presented by Barabasi, Albert and Jeong [1,5] and heuristic arguments given by Newman, Strogatz and Watts [23] suggest that after n steps the resulting graph should have diameter approximately logn. We show that while this holds for m=1, for m≥2 the diameter is asymptotically log n/log logn.

Robustness and Vulnerability of Scale-Free Random Graphs

Abstract: Recently many new "scale-free" random graph models have been introduced, motivated by the power-law degree sequences observed in many large-scale, real-world networks. Perhaps the best known, the Barabasi-Albert model, has been extensively studied from heuristic and experimental points of view. Here we consider mathematically two basic characteristics of a precise version of this model, the LCD model, namely robustness to random damage, and vulnerability to malicious attack. We show that the LCD graph is much more robust than classical random graphs with the same number of edges, but also more vulnerable to attack. In particular, if vertices of the n-vertex LCD graph are deleted at random, then as long as any positive proportion remains, the graph induced on the remaining vertices has a component of order n. In contrast, if the deleted vertices are chosen maliciously, a constant fraction less then 1 can be deleted to destroy all large components. For the Barabasi-Albert model, these questions have been st...

A multi-dimensional approach to force-directed layouts of large graphs

Abstract: We present a novel hierarchical force-directed method for drawing large graphs. Given a graph G=(V.E), the algorithm produces an embedding for G in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E. Such projections typically result in drawings that are "smoother" and more symmetric than direct drawings in 2D and 3D. In order to obtain fast placement of the vertices of the graph our algorithm employs a multi-scale technique based on a maximal independent set filtration of vertices of the graph. While most existing force-directed algorithms begin with an initial random placement of all the vertices, our algorithm attempts to place vertices "intelligently", close to their final positions. Other notable features of our approach include a fast energy function minimization strategy and efficient memory management. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a 550 MHz Pentium PC.

Random Deletion in a Scale-Free Random Graph Process

Abstract: We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and deletes vertices randomly. At time t, with probability α1 > 0 we add a new vertex ut and m random edges incident with ut . The neighbours of ut are chosen with probability proportional to degree. With probability α -α1 ≥ 0 we add m random edges to existing vertices where the endpoints are chosen with probability proportional to degree. With probability 1-α-α0 we delete a random vertex, if there are vertices left to delete. With probability α0 we delete m random edges. Assuming that α + α1 + α0 > 1 and α0 is sufficently small, we show that for large k, t, the expected number of vertices of degree k is approximately dkt where as k → 8, dk ~ Ck -1-β where and C > 0 is a constant. Note that β can take any value greater than 1.

Resistance matrix of a weighted graph

Abstract: In contrast to the classical notion of distance as the length of a shortest path between two vertices, the concept of resistance distance, introduced by Klein and Randic, arises naturally from several different considerations and is more amenable, to mathematical treatment. For a connected graph with n vertices, the resistance matrix of the graph is defined to be the n × n matrix with its (i, j)-entry equal to the resistance distance between the i-th and the j-th vertices. We obtain a formula for the inverse and the determinant of the resistance matrix of a weighted graph, thereby generalizing some earlier work, including that of Graham, Pollack, Lova'sz, Xiao and Gutman.

Bipartite anti-Ramsey numbers of cycles

Abstract: We determine the maximum number of colors in a coloring of the edges of Km,n such that every cycle of length 2k contains at least two edges of the same color. One of our main tools is a result on generalized path covers in balanced bipartite graphs. For positive integers q≤ a, let g(a,q) be the maximum number of edges in a spanning subgraph G of Ka,a such that the minimum number of vertex-disjoint even paths and pairs of vertices from distinct partite sets needed to cover V(G) is q. We prove that g(a,q) = a2 - aq + max {a, 2q - 2}. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 928, 2004 Part of the research of Tao Jiang was done at Michigan Technological University, Houghton, MI 49931.

Partitioning a graph in alliances and its application to data clustering

Abstract: Any reasonably large group of individuals, families, states, and parties exhibits the phenomenon of subgroup formations within the group such that the members of each group have a strong connection or bonding between each other. The reasons of the formation of these subgroups that we call alliances differ in different situations, such as, kinship and friendship (in the case of individuals), common economic interests (for both individuals and states), common political interests, and geographical proximity. This structure of alliances is not only prevalent in social networks, but it is also an important characteristic of similarity networks of natural and unnatural objects. (A similarity network defines the links between two objects based on their similarities). Discovery of such structure in a data set is called clustering or unsupervised learning and the ability to do it, automatically is desirable for many applications in the areas of pattern recognition, computer vision, artificial intelligence, behavioral and social sciences, life sciences, earth sciences, medicine, and information theory. In this dissertation, we study a graph theoretical model of alliances where an alliance of the vertices of a graph is a set of vertices in the graph, such that every vertex in the set is adjacent to equal or more vertices inside the set than the vertices outside it. We study the problem of partitioning a graph into alliances and identify classes of graphs that have such a partition. We present results on the relationship between the existence of such a partition and other well known graph parameters, such as connectivity, subgraph structure, and degrees of vertices. We also present results on the computational complexity of finding such a partition. An alliance cover set is a set of vertices in a graph that contains at least one vertex from every alliance of the graph. The complement of an alliance cover set is an alliance free set, that is, a set that does not contain any alliance as a subset. We study the properties of these sets and present tight bounds on their cardinalities. In addition, we also characterize the graphs that can be partitioned into alliance free and alliance cover sets. Finally, we present an approximate algorithm to discover alliances in a given graph. At each step, the algorithm finds a partition of the vertices into two alliances such that the alliances are strongest among all such partitions. The strength of an alliance is defined as a real number p, such that every vertex in the alliance has at least p times more neighbors in the set than its total number of neighbors in the graph). We evaluate the performance of the proposed algorithm on standard data sets.

Hole and antihole detection in graphs

Abstract: In this paper, we study the problems of detecting holes and antiholes in general undirected graphs and present algorithms for them, which, for a graph on n vertices and m edges, run in O(n + m2) time and require O(nm) space; we thus provide a solution to the open problem posed by Hayward, Spinrad, and Sritharan in [12] asking for an O(n4)-time algorithm for finding holes in arbitrary graphs. The key element of the algorithms is a special type of depth-first search traversal which proceeds along P4s (i.e., chordless paths on four vertices) of the input graph. We also describe a different approach which allows us to detect antiholes in graphs that do not contain chordless cycles on 5 vertices in O(n + m2) time requiring O(n + m) space. Our algorithms are simple and can be easily used in practice. Additionally, we show how our detection algorithms can be augmented so that they return a hole or an antihole whenever such a structure is detected in the input graph; the augmentation takes O(n + m) time and space.

Compact Mathematical Formulation for Graph Partitioning

Abstract: The graph partitioning problem consists of dividing the vertices of a graph into clusters, such that the weight of the edges crossing between clusters is minimized We present a new compact mathematical formulation of this problem, based on the use of binary representation for the index of clusters assigned to vertices This new formulation is almost minimal in terms of the number of variables and constraints and of the density of the constraint matrix Its linear relaxation brings a very fast computational resolution, compared with the standard one

On the Value of a Random Minimum Weight Steiner Tree

Abstract: Consider a complete graph on n vertices with edge weights chosen randomly and independently from an exponential distribution with parameter 1. Fix k vertices and consider the minimum weight Steiner tree which contains these vertices. We prove that with high probability the weight of this tree is (1+o(1))(k-1)(log n-log k)/n when k =o(n) and n→∞.

Many-to-Many disjoint path covers in a graph with faulty elements

Abstract: In a graph G, k vertex disjoint paths joining k distinct source-sink pairs that cover all the vertices in the graph are called a many-to-many k-disjoint path cover(k-DPC) of G We consider an f-fault k-DPC problem that is concerned with finding many-to-many k-DPC in the presence of f or less faulty vertices and/or edges We consider the graph obtained by merging two graphs H0 and H1, |V(H0)| = |V(H1)| = n, with n pairwise nonadjacent edges joining vertices in H0 and vertices in H1 We present sufficient conditions for such a graph to have an f-fault k-DPC and give the construction schemes Applying our main result to interconnection graphs, we observe that when there are f or less faulty elements, all of recursive circulant G(2m,4), twisted cube TQm, and crossed cube CQm of degree m have f-fault k-DPC for any k ≥ 1 and f ≥ 0 such that f + 2k ≤ m–1.

Fast tree-based generation of a dependence graph

Abstract: A dependence graph having a linear number of edges and one or more tie vertices is generated by constructing a tree of nodes, receiving requests to create cut and/or fan vertices corresponding to each node, adjusting a frontier of nodes up or down, and creating one or more cut or fan vertices, zero or more tie vertices, and at least one predecessor edge.

Chordless Paths Through Three Vertices

Abstract: Consider the following problem, that we call “Chordless Path through Three Vertices” or Cp3v, for short: Given a simple undirected graph G=(V,E), a positive integer k, and three distinct vertices s, t, and v∈ V, is there a chordless path from s via v to t in G that consists of at most k vertices? In a chordless path, no two vertices are connected by an edge that is not in the path. Alternatively, one could say that the subgraph induced by the vertex set of the path in G is the path itself. The problem has been raised in the context of service deployment in communication networks. We resolve the parametric complexity of Cp3v by proving it W[1]-complete with respect to its natural parameter k. Our reduction extends to a number of related problems about chordless paths. In particular, deciding on the existence of a single directed chordless (s,t)-path in a digraph is also W[1]-complete with respect to the length of the path.

Minsquare Factors and Maxfix Covers of Graphs

Abstract: We provide a polynomial algorithm that determines for any given undirected graph, positive integer k and various objective functions on the edges or on the degree sequences, as input, k edges that minimize the given objective function. The tractable objective functions include linear, sum of squares, etc. The source of our motivation and at the same time our main application is a subset of k vertices in a line graph, that cover as many edges as possible (maxfix cover). Besides the general algorithm and connections to other problems, the extension of the usual improving paths for graph factors could be interesting in itself: the objects that take the role of the improving walks for b-matchings or other general factorization problems turn out to be edge-disjoint unions of pairs of alternating walks. The algorithm we suggest also works if for any subset of vertices upper, lower bound constraints or parity constraints are given. In particular maximum (or minimum) weight b-matchings of given size can be determined in polynomial time, combinatorially, in more than one way.

On locally Delaunay geometric graphs

Abstract: A geometric graph is a simple graph G=(V,E) with an embedding of the set V in the plane such that the points that represent V are in general position. A geometric graph is said to be k-locally Delaunay (or a Dk-graph) if for each edge (u,v) ∈ E there is a (Euclidean) disc d that contains u and v but no other vertex of G that is within k hops from u or v.The study of these graphs was recently motivated by topology control for wireless networks [6,7]. We obtain the following results: (i) We prove that if G is a D1-graph on n vertices, then it has O(n3/2) edges. (ii) We show that for any n there exist D1-graphs with n vertices and Ω(n4/3) edges. (iii) We prove that if G is a D2-graph on n vertices, then it has O(n) edges. This bound is worst-case asymptotically tight. As an application of the first result, we show that: (iv) The maximum size of a family of pairwise non-overlapping lenses in an arrangement of $n$ unit circles in the plane is O(n3/2).The first two results improve the best previously known upper and lower bounds of $O(n^ 5/3 )$ and $\Omega(n)$ respectively (see \cite KL03 ). The third result improves the best previously known upper bound of O(n log n ) ([6]). Finally, our last result improves the best previously known upper bound (for the more general case of not necessarily unit circles) of O(n3/2 κ(n)) (see [1] ), where κ(n) = (log n ) O(α2(n)) and where α(n) is the extremely slowly growing inverse Ackermann's function.

Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality

Abstract: In this paper we study non-interactive correlation distillation (NICD), a generalization of the study of noise sensitivity of boolean functions. We extend the model to NICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating. Our new contributions include the following: 1. In the case of a k-leaf star graph, we resolve the open question of whether the success probability must go to zero as k goes to infinity. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowly-decaying polynomial). 2. In the case of the k-vertex path graph, we show that it is always optimal for all players to use the same 1-bit function. 3. In the general case we show that all players should use monotone functions. 4. For certain trees it is better if not all players use the same function. Our techniques include the use of the reverse Bonami-Beckner inequality.

Drawing Planar Graphs with Large Vertices and Thick Edges

Abstract: We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and information traveling through the network. We present an efficient linear-time algorithm which draws edges and vertices of varying 2-dimensional areas to represent the amount of information flowing through them. The algorithm avoids all occlusions of nodes and edges, while still drawing the graph on a compact integer grid.

Generalizing Pancyclic and k -Ordered Graphs

Abstract: Given positive integers k≤m≤n, a graph G of order n is (k,m)-pancyclic if for any set of k vertices of G and any integer r with m≤r≤n, there is a cycle of length r containing the k vertices Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k,m)-pancylic are proved If the additional property that the k vertices must appear on the cycle in a specified order is required, then the graph is said to be (k,m)-pancyclic ordered Minimum degree conditions and minimum sum of degree conditions for nonadjacent vertices that imply a graph is (k,m)-pancylic ordered are also proved Examples showing that these constraints are best possible are provided

System and method for diffusing curvature

Abstract: A system or method to distribute curvature in a set of target vertices by computing curvature at boundary vertices of the set of target vertices by use of an umbrella operator. The boundary curvatures may be distributed into the set of target vertices by solving for a system of Umbrella operator equations for curvatures of respective vertices of the set of target vertices, with the computed curvature at the boundary vertices as a boundary condition for the system of equations. The vertices of the set of target vertices may be repositioned relative to the their neighbors according to the solved curvatures of the respective vertices of the set of vertices. The computing, distributing, and repositioning may be repeated, thereby changing the overall shape of the set of target vertices according to the curvature at the boundary vertices.

Edge-disjoint routing in plane switch graphs in linear time

Abstract: By a switch graph, we mean an undirected graph G = (P c W, E) such that all vertices in P (the plugs) have degree one and all vertices in W (the switches) have even degrees. We call Gplane if G is planar and can be embedded such that all plugs are in the outer face. Given a set (s1, t1), …,(sk, tk) of pairs of plugs, the problem is to find edge-disjoint paths p1, …, pk such that every pi connects si with ti. The best asymptotic worst-case complexity known so far is quadratic in the number of vertices. In this article, a linear, and thus asymptotically optimal, algorithm is introduced. This result may be viewed as a concluding "keystone" for a number of previous results on various special cases of the problem.

Parameterized Graph Separation Problems

Abstract: We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given l terminals is separated from the others, (b) each of the given l pairs of terminals are separated, (c) exactly l vertices are cut away from the graph, (d) exactly l connected vertices are cut away from the graph, (e) the graph is separated into l components, We show that if both k and l are parameters, then (a), (b) and (d) are fixed-parameter tractable, while (c) and (e) are W[1]-hard.