Showing papers on "Path graph published in 2003"

An Ideal-Based Zero-Divisor Graph of a Commutative Ring

Abstract: For a commutative ring R with identity, the zero-divisor graph of R, denoted Γ(R), is the graph whose vertices are the non-zero zero-divisors of R with two distinct vertices joined by an edge when the product of the vertices is zero. We will generalize this notion by replacing elements whose product is zero with elements whose product lies in some ideal I of R. Also, we determine (up to isomorphism) all rings R such that Γ(R) is the graph on five vertices.

A fast path planning by path graph optimization

Abstract: A fast path planning method by optimization of a path graph for both efficiency and accuracy is proposed. A conventional quadtree-based path planning approach is simple, robust, and efficient. However, it has two limitations. We propose a path graph optimization technique employing a compact mesh representation. A world space is triangulated into a base mesh and the base mesh is simplified to a compact mesh. The compact mesh representation is object-dependent; the positions of vertexes of the mesh are optimized according to the curvatures of the obstacles. The compact mesh represents the obstacles as accurately as the quadtree even though using much fewer vertexes than the quadtree. The compact mesh distributes vertexes in a free space in a balanced way by ensuring that the lengths of edges are below an edge length threshold. An optimized path graph is extracted from the compact mesh. An iterative vertex pushing method is proposed to include important obstacle boundary edges in the path graph. Dijkstra's shortest path searching algorithm is used to search the shortest path in the path graph. Experimental results show that the path planning using the optimized path graph is an order of magnitude faster than the quadtree approach while the length of the path generated by the proposed method is almost the same as that of the path generated by the quadtree.

Ramsey Games Against a One-Armed Bandit

Abstract: We study the following one-person game against a random graph process: the Player's goal is to $2$-colour a random sequence of edges $e_1,e_2,\dots$ of a complete graph on $n$ vertices, avoiding a monochromatic triangle for as long as possible The game is over when a monochromatic triangle is created The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edgeWhile it is not hard to prove that the expected length of this game is about $n^{4/3}$, the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with $cn^{3/2}$ edges, where $c$ is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round

Application of Chemical Graph Theory for Automated Mechanism Generation

Abstract: We present an application of the chemical graph theory approach for generating elementary reactions of complex systems. Molecular species are naturally represented by graphs, which are identified by their vertices and edges where vertices are atom types and edges are bonds. The mechanism is generated using a set of reaction patterns (sub-graphs). These subgraphs are the internal representations for a given class of reaction thus allowing for the possibility of eliminating unimportant product species a priori. Furthermore, each molecule is canonically represented by a set of topological indices (Connectivity Index, Balaban Index, Schulz TI Index, WID Index, etc.) and thus eliminates the probability for regenerating the same species twice. Theoretical background and test cases on combustion of hydrocarbons are presented.

Finding a Path of Superlogarithmic Length

Abstract: We consider the problem of finding a long, simple path in an undirected graph. We present a polynomial-time algorithm that finds a path of length $\Omega\bigl((\log L/\log\log L)^2\bigr)$, where L denotes the length of the longest simple path in the graph. This establishes the performance ratio O(n(log log n/log n)2) for the longest path problem, where n denotes the number of vertices in the graph.

Worst-case-optimal algorithms for guarding planar graphs and polyhedral surfaces

Abstract: We present an optimal Θ(n)-time algorithm for the selection of a subset of the vertices of an n-vertex plane graph G so that each of the faces of G is covered by (i.e., incident with) one or more of the selected vertices. At most ⌊n/2⌋ vertices are selected, matching the worst-case requirement. Analogous results for edge-covers are developed for two different notions of "coverage". In particular, our linear-time algorithm selects at most n - 2 edges to strongly cover G, at most ⌊n/3⌋ diagonals to cover G, and in the case where G has no quadrilateral faces, at most ⌊n/3⌋ edges to cover G. All these bounds are optimal in the worst-case. Most of our results flow from the study of a relaxation of the familiar notion of a 2-coloring of a plane graph which we call a face-respecting 2-coloring that permits monochromatic edges as long as there are no monochromatic faces. Our algorithms apply directly to the location of guards, utilities or illumination sources on the vertices or edges of polyhedral terrains, polyhedral surfaces, or planar subdivisions.

Impact of local topological information on random walks on finite graphs

Abstract: It is just amazing that both of the mean hitting time and the cover time of a random walk on a finite graph, in which the vertex visited next is selected from the adjacent vertices at random with the same probability, are bounded by O(n3) for any undirected graph with order n, despite of the lack of global topological information. Thus a natural guess is that a better transition matrix is designable if more topological information is available. For any undirected connected graph G = (V,E), let P(β) = (puvβ)u,v∈V be a transition matrix defined by puvβ = exp [-βU(u, v)]/Σw∈N(u) exp [-βU(u, w)] for u∈V, v∈N(u), where β is a real number, N(u) is the set of vertices adjacent to a vertex u, deg(u) = |N(u)|, and U(., .) is a potential function defined as U(u, v) = log (max {deg(u), deg(v)}) for u∈V, v∈N(u). In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time with respect to P(1) are bounded by O(n2 log n) and O(n2), respectively. We further show that P(1) is best possible with respect to the mean hitting time, in the sense that the mean hitting time of a path graph of order n, with respect to any transition matrix, is Ω(n2).

Disjoint edges in topological graphs

Abstract: A topological graph G is a graph drawn in the plane so that its edges are represented by Jordan arcs. G is called simple, if any two edges have at most one point in common. It is shown that the maximum number of edges of a simple topological graph with n vertices and no k pairwise disjoint edges is O(nlog4k−8n) edges. The assumption that G is simple cannot be dropped: for every n, there exists a complete topological graph of n vertices, whose any two edges cross at most twice.

Apparatus and methods for analyzing graphs

Abstract: A plurality of hardware cells are defined, wherein at least a given one of the hardware cells corresponds to sets of vertices from a graph having vertices and edges interconnecting the vertices, and each of the sets are from a corresponding one of a number of portions of the graph. The given hardware cell is adapted to select one of the sets of vertices and to define for the selected set of vertices whether an edge exists in the graph between the vertices in the selected set. The hardware cells are used to analyze one or more properties of the graph, such as reachability or shortest path. The graph is mapped into an adjacency matrix, which contains a number of contexts, each context having a number of elements, and where the given hardware cell corresponds to multiple contexts of the adjacency matrix.

Graph Connectivity After Path Removal

Abstract: Let G be a graph and u, v be two distinct vertices of G A u—v path P is called nonseparating if G—V(P) is connected The purpose of this paper is to study the number of nonseparating u—v path for two arbitrary vertices u and v of a given graph For a positive integer k, we will show that there is a minimum integer α(k) so that if G is an α(k)-connected graph and u and v are two arbitrary vertices in G, then there exist k vertex disjoint paths P1[u,v], P2[u,v], , Pk[u,v], such that G—V (Pi[u,v]) is connected for every i (i = 1, 2, , k) In fact, we will prove that α(k) ≤ 22k+2 It is known that α(1) = 3 A result of Tutte showed that α(2) = 3 We show that α(3) = 6 In addition, we prove that if G is a 5-connected graph, then for every pair of vertices u and v there exists a path P[u, v] such that G—V(P[u, v]) is 2-connected

Ch. 2. Models of random graphs and their applications

Abstract: Publisher Summary Networks are ubiquitous. They arise naturally as models of communication networks, networks of friends, in the communication of infection, rumors or information, as models of atoms and bonds between them in chemistry, as autocatalytic nets and elsewhere. Mathematically the notion is captured in a graph: a finite set of vertices V and a set E of edges between some of the distinct vertices. This chapter presents graphs that have finite vertex set; do not have multiple edges between two vertices or loops from a vertex to itself, and whose edges are undirected. The chapter introduces Erdős–Renyi model; a natural generalization of Erdős–Renyi model random graphs is made when the edge between vertices v1 and v2 arises with probability p v1v2 , independently of all other edges. Although the Erdős–Renyi model is mathematically tractable; there is mathematical interest in comparing it with alternative models. Additionally, in many real networks, edges will not in fact arise independently and equiprobably.

A Characterization of Planar Graphs by Pseudo-Line Arrangements

Abstract: Let 𝜞 be an arrangement of pseudo-lines, i.e., a collection of unbounded x -monotone curves in which each curve crosses each of the others exactly once. A pseudo-line graph (𝜞, E) is a graph for which the vertices are the pseudo-lines of 𝜞 and the edges are some unordered pairs of pseudo-lines of 𝜞. A diamond of a pseudo-line graph (𝜞, E) is a pair of edges {p,q} , {p',q'}∈ E , {p,q} ∩ {p',q'}= O, such that the crossing point of the pseudo-lines p and q lies vertically between p' and q' and the crossing point of p' and q' lies vertically between p and q . We show that a graph is planar if and only if it is isomorphic to a diamond-free pseudo-line graph. An immediate consequence of this theorem is that the O(k1/3n) upper bound on the k -level complexity of an arrangement of straight lines, which was very recently discovered by Dey, holds for an arrangement of pseudo-lines as well.

On the Structure of a k-Connected Graph

Abstract: For a k-connected graph G, we introduce the notion of a block and construct a block tree. This construction generalizes, for \(k \geqslant 1\), the known constructions for blocks of a connected graph. We apply the introduced notions to describe the set of vertices of a k-connected graph G such that the graph remains k-connected after deleting these vertices. We discuss some problems related to simultaneous deleting of vertices of a k-connected graph without loss of k-connectivity. Bibliography: 5 titles.

Topological graphs with no self-intersecting cycle of length 4

Abstract: Let G be a topological graph on n vertices in the plane, i.e., a graph drawn in the plane with its vertices represented as points and its edges represented as Jordan arcs connecting pairs of points. It is shown that if no two edges of any cycle of length 4 in G cross an odd number of times, then |E(G)|=O(n8/5).

Antiregular graphs are universal for trees

Abstract: A graph on n vertices is antiregular if its vertex degrees take on n − 1 different values. For every n ≥ 2 there is a unique connected antiregular graph on n vertices. Call it An. (The unique disconnected antiregular graph on n vertices is Acn.) The main result of this note is that every tree on n vertices is isomorphic to a subgraph of An.

Improved shortest path algorithms for nearly acyclic graphs

Abstract: Dijkstra's algorithm solves the single-source shortest path problem on any directed graph in O(m + n log n) time when a Fibonacci heap is used as the frontier set data structure. Here n is the number of vertices and m is the number of edges in the graph. If the graph is nearly acyclic, other algorithms can achieve a time complexity lower than that of Dijkstra's algorithm. Abuaiadh and Kingston gave a single-source shortest path algorithm for nearly acyclic graphs with O(m + n log t) time complexity, where the new parameter, t, is the number of delete-min operations performed in priority queue manipulation. If the graph is nearly acyclic, then t is expected to be small, and the algorithm out-performs Dijkstra's algorithm. Takaoka, using a different definition for acyclicity, gave an algorithm with O(m + n log k) time complexity. In this algorithm, the new parameter, k, is the maximum cardinality of the strongly connected components in the graph.The generalised single-source (GSS) problem allows an initial distance to be defined at each vertex in the graph. Decomposing a graph into r trees allows the GSS problem to be solved within O(m + r logr) time. This paper presents a new all-pairs algorithm with a time complexity of O(mn + nr log r), where r is the number of acyclic parts resulting when the graph is decomposed into acyclic parts. The acyclic decomposition used is setwise unique and can be computed in O(mn) time. If the decomposition has been pre-calculated, then GSS can be solved within O(m + r log r) time whenever edge-costs in the graph change. A second new all-pairs algorithm is presented, with O(mn + nr2) worst-case time complexity, where r is the number of vertices in a pre-calculated feedback vertex set for the nearly acyclic graph. For certain graphs, these new algorithms offer an improvement on the time complexity of the previous algorithms.

An Efficient Algorithm for Finding All Hinge Vertices on Trapezoid Graphs

Abstract: The distance between at least two vertices becomes longer after the removal of a hinge vertex. Thus a faulty hinge vertex will increase the overall communication cost in a network. Therefore, finding the set of all hinge vertices in a graph can be used to identify critical nodes in a real network. An O(n log n) time algorithm has been proposed here to find all hinge vertices of a trapezoid graph with n vertices.

4-valent plane graphs with 2-, 3- and 4-gonal faces

Abstract: Call i-hedrite any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only and p2+p3 = i. The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its central circuits, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge opposite the entering one. So, any i-hedrite is a projection of an alternating link, whose components correspond to its central circuits. Call an i-hedrite irreducible, if it has no rail-road, i.e. a circuit of 4-gonal faces, in which every 4-gon is adjacent to two of its neighbors on opposite edges. We present the list of all i-hedrites with at most 15 vertices. Examples of other results: (i) All i-hedrites, which are not 3-connected, are identified. (ii) Any irreducible i-hedrite has at most i − 2 central circuits. (iii) All i-hedrites without self-intersecting central circuits are listed. (iv) All symmetry group of i-hedrites are listed.

Noncrossing Hamiltonian paths in geometric graphs

Abstract: A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: Determine a function h, where h(n) is the largest number k such that when we remove arbitrary set of k edges from a complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that h(n) = Ω(√n). We also determine the function exactly in case when the removed edges form a star or a matching, and give asymptotically tight bounds in case they form a clique.

Finding large independent sets in polynomial expected time

Abstract: We consider instances of the maximum independent set problem that are constructed according to the following semirandom model. First, let G n,p be a random graph, and let S be a set consisting of k vertices, chosen uniformly at random. Then, let Go be the graph obtained by deleting all edges connecting two vertices in S. Adding to Go further edges that do not connect two vertices in S, an adversary completes the instance G = G* n,p,k We propose an algorithm that in the case k > C(n/p) 1/2 on input G within polynomial expected time finds an independent set of size > k.

Representing Graph Metrics with Fewest Edges

Abstract: We are given a graph with edge weights, that represents the metric on the vertices in which the distance between two vertices is the total weight of the lowest-weight path between them. Consider the problem of representing this metric using as few edges as possible, provided that new "steiner" vertices (and edges incident on them) can be added. The compression factor achieved is the ratio k between the number of edges in the original graph and the number of edges in the compressed graph. We obtain approximation algorithms for unit weight graphs that replace cliques with stars in cases where the cliques so compressed are disjoint, or when only a constant number of the cliques compressed meet at any vertex. We also show that the general unit weight problem is essentially as hard to approximate as graph coloring and maximum clique.