Showing papers on "Path graph published in 1996"

Graphs Drawn with Few Crossings Per Edge

Abstract: We show that if a graph of v vertices can be drawn in the plane so that every edge crosses at most k> 0 others, then its number of edges cannot exceed 4.108√kv. For k≤ 4, we establish a better bound, (k + 3)(u− 2), which is tight for k=1 and 2. We apply these estimates to improve a result of Ajtai et al. and Leighton, providing a general lower bound for the crossing number of a graph in terms of its number of vertices and edges.

On Linear Recognition of Tree-Width at Most Four

Abstract: A graph $G$ has tree-width at most $k$ if the vertices of $G$ can be decomposed into a tree-like structure of sets of vertices, each set having cardinality at most $k+1$. An alternate definition of tree-width is stated in terms of a $k$-elimination sequence, which is an order to eliminate the vertices of the graph such that each vertex, at the time it is eliminated from the graph, has degree at most $k$. Arnborg and Proskurowski showed that if a graph has tree-width at most a fixed $k$, then many NP-hard problems can be solved in linear time, provided this $k$-elimination sequence is part of the input. These algorithms are very efficient for small $k$, such as 2, 3, or 4, but may be impractical for large $k$ as they depend exponentially on $k$. A reduction process is developed, and reductions are shown that can be applied to a graph of tree-width at most four without increasing its tree-width. Further, each graph of tree-width at most four contains one of these reductions. The reductions are then used in a linear-time algorithm that generates a 4-elimination sequence, if one exists.

Random Walks on Regular and Irregular Graphs

Abstract: For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list until it completes a full cycle. The main result of this paper is a characterization of the cyclic cover time in terms of simple and easy-to-compute graph properties. Namely, for any connected graph, the cyclic cover time is $\Theta(n^2 d_{ave} (d^{-1})_{ave})$, where $n$ is the number of vertices in the graph, $d_{ave}$ is the average degree of its vertices, and $(d^{-1})_{ave}$ is the average of the inverse of the degree of its vertices. Other results obtained in the processes of proving the main theorem are a similar characterization of minimum resistance spanning trees of graphs, improved bounds on the cover time of graphs, and a simplified proof that the maximum commute time in any connected graph is at most $4n^3/27 + o(n^3)$.

The number of edge colorings with no monochromatic triangle

Abstract: Let F (n, k) denote the maximum number of two edge colorings of a graph on n vertices that admit no monochromatic Kk, (a complete graph on k vertices). The following results are proved: F (n, 3) = 2bn /4c for all n ≥ 6. F (n, k) = 2 k−2 2k−2+o(1))n 2 . In particular, the first result solves a conjecture of Erdos and Rothschild.

Network observability: theory and algorithms based on triangular factorisation and path graph concepts

Abstract: A theory is developed to test network observability, to identify possible subnetworks which might be observable, and, in case the network is not observable, to identify the minimal set of measurements (pseudo) to be added so that the network as a whole becomes observable Algorithms to test network observability and to identify subnetworks of the entire system that might be observable are also presented An algorithm to add a minimal set of measurements to the existing measurements, so that the network becomes observable as a whole, is developed The theory and algorithms that result are a combination of factorisation path concepts and triangularisation of the gain matrix G Such algorithms have the characteristics of being very simple, easy to implement, extremely fast and reliable Algorithms resulting from this theory do not require solutions of any algebraic equation

On distances in benzenoid systems

Abstract: We show that the molecular graph G of a benzenoid hydrocarbon admits an isometric embedding into the Cartesian product of three trees T1, T2, and T3 defined by three directions of the host hexagonal grid. Namely, to every vertex v of G one can associate an ordered triplet (v1, v2, v3) with vi being a vertex of Ti (i = 1, 2, 3), such that the graph-theoretic distance between two vertices u, v of G equals the sum of respective tree-distances between ui and vi. This labeling of the vertices of G can be obtained in O(n) time. As an application of this result we present an optimal O(n) time algorithm for computing the diameter of the graph G of a benzenoid system with n vertices.

The Linkage of a Graph

Abstract: The linkage of a graph is defined to be the maximum min-degree of any of its subgraphs. It is known that the linkage of a graph is equal to its width: for an arbitrary linear ordering of the vertices of the graph, consider the maximum, with respect to any vertex $v$, of the number of vertices connected with $v$ and preceding it in the ordering; the width of the graph is the minimum of these maxima over all possible linear orderings. Width has been used in artificial intelligence in the context of constraint satisfaction problems (CSPs). A more general notion is defined by considering not the number of vertices preceding and connected with $v$ but rather the least number of vertices preceding and connected with any cluster of at most $j$ consecutive vertices extending to the right up to $v$ ($j$ is a given integer). The graph parameter thus defined is called $j$-width. No efficient algorithm was known for computing the $j$-width. In this paper, we introduce a graph parameter depending on $j$ that refers to the subgraphs of the graph and generalizes the notion of linkage. We prove the min--max theorem that this graph parameter, which we call $j$-linkage, is equal to $j$-width, and we then give a polynomial-time algorithm for computing it (for constant $j$). We also find tight lower and upper bounds for the $j$-linkage (equivalently, the $j$-width) of graphs with given numbers of vertices and edges. It is interesting to note that a lower bound for the width of a graph had been found by Erdos; as we show, however, that bound is not tight. Moreover, we prove that our lower bound for width is also a tight lower bound for treewidth, pathwidth, and bandwidth, graph parameters that may be arbitrarily larger than width. Finally, we show that computing the $j$-linkage is a P-complete problem, whereas we prove that approximating it is a threshold problem: it is in NC for approximation factors $ 1/2$.

An efficient algorithm for the vertex-disjoint paths problem in random graphs

Abstract: Given a graph G = (V, E) and a set of pairs of vertices in V, we are interested in finding for each pair (a{sub i}, b{sub i}) a path connecting a{sub i} to b{sub i}, such that the set of paths so found is vertex-disjoint. (The problem is NP-complete for general graphs as well as for planar graphs. It is in P if the number of pairs is fixed.) Our model is that the graph is chosen first, then an adversary chooses the pairs of endpoints, subject only to obvious feasibility constraints, namely, all pairs must be disjoint, no more than a constant fraction of the vertices could be required for the paths, and not {open_quotes}too many{close_quotes} neighbors of a vertex can be endpoints. We present a randomized polynomial time algorithm that works for almost all graphs; more precisely in the G{sub n,m} or G{sub n,p} models, the algorithm succeeds with high probability for all edge densities above the connectivity threshold. The set of pairs that can be accommodated is optimal up to constant factors. Although the analysis is intricate, the algorithm itself is quite simple and suggests a practical heuristic. We include two applications of the main result,more » one in the context of circuit switching communication, the other in the context of topological embeddings of graphs.« less

The wide-diameter of the n-dimensional toroidal mesh

Abstract: In graph theory and a study of transmission delay and fault tolerance of networks, the connectivity and the diameter of a graph are very important and they have been studied by many mathematicians. We studied the wide-diameter of a k-regular k-connected graph which is defined by the maximum of the k-distance between two distinct vertices, when the k-distance between x and y is equal to the least number l such that there exists k vertex-disjoint paths between x and y whose lengths are at most l. Because the wide-diameter of any k-regular k-connected graph is greater than its diameter, a k-regular k-connected graph whose wide-diameter is equal to “1 + its diameter” is optimal. For example, it is known that the hypercube is such a graph. We define n-dimensional toroidal mesh C(d1, d2, ………, dn) with vertices {(x1, ………, xn)|0 ≤ xi < di (1 ≤ i ≤ n)}. Each vertex (x1, ………, xn) is adjacent to 2n other vertices: (x1 ± 1, x2, ………, xn), (x1, x2 ± 1, ………, xn), ………, (x1, x2, ………, xn ± 1), where additions are performed modulo di (1 ≤ i ≤ n). This graph is an n-dimensional orthogonal mesh with global edges, which is 2n-connected and 2n-regular. We show that the graph satisfies the above property with respect to its 2n-diameter and its diameter except for some special cases. © 1996 John Wiley & Sons, Inc.

On k -saturated graphs with restrictions on the degrees

Abstract: A graph G is called k-saturated, where k ≥ 3 is an integer, if G is K-free but the addition of any edge produces a K (we denote by K a complete graph on k vertices). We investigate k-saturated graphs, and in particular the function Fk(n,D) defined as the minimal number of edges in a k-saturated graph on n vertices having maximal degree at most D. This investigation was suggested by Hajnal, and the case k = 3 was studied by Furedi and Seress. The following are some of our results. For k = 4, we prove that F4(n,D) = 4n− 15 for n > n0 and ⌊ 2n−1 3 ⌋ ≤ D ≤ n− 2. For arbitrary k, we show that the limit limn→∞ Fk(n, cn)/n exists for all 0 < c ≤ 1, except maybe for some values of c contained in a sequence ci → 0. We also determine the asymptotic behaviour of this limit for c → 0. We construct, for all k and all sufficiently large n, a k-saturated graph on n vertices with maximal degree at most 2k √ n, significantly improving an upper bound due to Hanson and Seyffarth. ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Research supported in part by the Fund for Basic Research administered by the Israel Academy of Sciences. †Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary, and Department of Mathematics, Technion Israel Institute of Technology, Haifa, Israel. ‡Department of Mathematics, Technion Israel Institute of Technology, Haifa, Israel. Research supported by the Tragovnik research fund and by the fund for the promotion of research at the Technion. Corresponding author. §Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. Research supported in part by a Charles Clore Fellowship.

A simple O(n2) algorithm for the all-pairs shortest path problem on an interval graph

Abstract: Let G denote an interval graph with n vertices and unit weight edges. In this paper, we present a simple O(n 2 ) algorithm for solving the all-pairs shortest path problem on graph G. A recent algorithm for this problem has the same time-complexity but is fairly complicated to describe. However, our algorithm is concise to state, intuitive to understand, and easy to implement.

On Extremal Graphs With No Long Paths

Abstract: Connected graphs with minimum degree ‐ and at least 2‐ + 1 vertices have paths with at least 2‐ + 1 vertices. We provide a characterization of all such graphs which have no longer paths.

Optimal routings in communication networks with linearly bounded forwarding index

Abstract: In a given graph with n vertices, a routing is defined as a set of n(n - 1) routes, one route connecting each ordered pair of vertices. The load of a vertex is the number of routes going through it. The forwarding index of the graph is the minimum of the largest load taken over all routings. We construct undirected graphs with a high degree of symmetry and specified diameter, in which the load of every vertex is at most constant times the number of vertices. This gives a partial solution to a problem of Chung et al.

Mapping in Random Structures

Abstract: A mapping in random structures is defined on the vertices of a generalized hypercube {\cal Q}^n_\alpha. A random structure consists of (i) a random contact graph and (ii) a family of relations inposed on adjacent vertices of the random contact graph. The vertex set of a random contact graph is the set of all coordinates of a vertex V \in {\cal Q}^n_\alpha, {P_1, \ldots, P_n}. The edges of the random contact graph are the union of the edge sets of two random graphs. The first is a random 1-regular graph on 2m vertices {p_{i_1}, \ldots, P_{i_{2m}}} and the second is a random graph G_p with p = {c_2\over n} on {P_1,\ldots,{_n}. The structure of the random contact graphs is investigated and it is shown that for certain values of m and c_2 the mapping in random structures allows systematic search in the set of random structures. Finally the results are applied to evolutionary optimization of biopolymers. Key words. random structure, sequence-structure mapping, random graph, connectivity, branching process, optimization

A lower bound for groupies in graphs

Abstract: A non-isolated vertex of a graph G is called a groupie if the average degree of the vertices connected to it is larger than or equal to the average degree of the vertices in G. An isolated vertex is a groupie only if all vertices of G are isolated. While it is well known that every graph must contain at least one groupie, the graph Kn − e contains just 2 groupie vertices for n ≥ 2. In this paper we derive a lower bound for the number of groupies which shows, in particular, that any graph with 2 or more vertices must contain at least 2 groupies. © 1996 John Wiley & Sons, Inc.

An algorithm for finding many disjoint monochromatic edges in a complete 2-colored geometric graph

Abstract: We present an $O(n^{\log\log n+2})$-time algorithm for finding $n$ disjoint monochromatic edges in a complete geometric graph of $3n-1$ vertices, where the edges are colored by two colors.

Directed path graphs

Abstract: The concept of a line digraph is generalized to that of a directed path graph. The directed path graph $\overrightarrow P_k(D)$ of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k + 1 vertices or form a directed cycle on k vertices in D. Several properties of $\overrightarrow P_k(D)$ are studied, in particular with respect to isomorphism and traversability.

Linear transport of particles on networks

Abstract: We introduce a model of transport of particles in a network, which is represented by a connected graph with m vertices and n edges. Each edge represents a one-dimensional conductor of particles, whose behavior is described by means of a linear Boltzmann-like equation. In graph vertices, a system of linear boundary conditions is given which takes into account the exchanges of particles between the edges. The well-posedness of the initial value problem is studied into an abstract L1-like setting and the structure of the solution is given for simplest case of pure streaming transport.