Showing papers on "Path graph published in 1988"

Rubber bands, convex embeddings and graph connectivity"

Abstract: We give various characterizations ofk-vertex connected graphs by geometric, algebraic, and “physical” properties. As an example, a graphG isk-connected if and only if, specifying anyk vertices ofG, the vertices ofG can be represented by points of ℝk−1 so that nok are on a hyper-plane and each vertex is in the convex hull of its neighbors, except for thek specified vertices. The proof of this theorem appeals to physics. The embedding is found by letting the edges of the graph behave like ideal springs and letting its vertices settle in equilibrium. As an algorithmic application of our results we give probabilistic (Monte-Carlo and Las Vegas) algorithms for computing the connectivity of a graph. Our algorithms are faster than the best known (deterministic) connectivity algorithms for allk≧√n, and for very dense graphs the Monte Carlo algorithm is faster by a linear factor.

On the complexity of covering vertices by faces in a planar graph

Abstract: The pair $(G,D)$ consisting of a planar graph $G = (V,E)$ with n vertices together with a subset of d special vertices $D \subseteq V$ is called k-planar if there is an embedding of G in the plane so that at most k faces of G are required to cover all of the vertices in D. Checking 1-planarity can be done in linear-time since it reduces to a problem of checking planarity of a related graph. We present an algorithm which given a graph G and a value k either determines that G is not k-planar or generates an appropriate embedding and associated minimum cover in $O(c^k n)$ time, where c is a constant. Hence, the algorithm runs in linear time for any fixed k. The fact that the time required by the algorithm grows exponentially in k is to be expected since we also show that for arbitrary k, the associated decision problem is strongly NP-complete, even when the planar graph has essentially a unique planar embedding, $d = \theta (n)$, and all facial cycles have bounded length. These results provide a polynomial-t...

Pagenumber of complete bipartite graphs

Abstract: Given an ordering of the vertices of a graph around a circle, a page is a collection of edges forming noncrossing chords. A book embedding is a circular permutation of the vertices together with a partition of the edges into pages. The pagenumber t(G) (also called book thickness) is the minimum number of pages in a book embedding of G. We present a general construction showing t(Km,n) ⩽ ⌈(m + 2n)/4⌉, which we conjecture optimal. We prove a result suggesting this is optimal for m ⩾ 2n − 3. For the most difficult case m = n, we consider vertex permutations that are regular, i.e., place vertices from each partite set into runs of equal size. Book embeddings with such orderings require ⌈(7n − 2)/9⌉ pages, which is achievable. The general construction uses fewer pages, but with an irregular ordering.

Some results on odd factors of graphs

Abstract: A {1, 3, …,2n − 1}-factor of a graph G is defined to be a spanning subgraph of G, each degree of whose vertices is one of {1, 3, …, 2n − 1}, where n is a positive integer. In this paper, we give a sufficient condition for a graph to have a {1, 3, …, 2n − 1}-factor.

An efficient ordering algorithm to improve sparse vector methods

Abstract: The authors present a novel node-ordering algorithm to enhance sparse vector methods in power system analysis. The proposed technique locally minimizes the number of nonzero elements of the inverse of the table of factors. It uses the cardinality of the set of nodes which precede each node in the path graph as a tie-break criterion in the minimum degree elimination process. Test results are included showing that the method performs better than previously published methods. >

Small topological complete subgraphs of dense graphs

Abstract: A graph ofn vertices and\(4^{t^2 } n^{1 + \varepsilon } \) edges contains aTKt on at most 7t2 logt/e vertices. This answers a question of P. Erdős.

A fast path-planning algorithm by synchronizing modification and search of its path graph (mobile robots)

Abstract: Determination of the shortest collision-free path for a mobile robot between start and goal positions in a workspace is central to the design of an autonomous mobile robot. The authors present a feasible path-planning algorithm which runs on the quadtree representation using a path graph. The quadtree representing the workspace is obtained from fast conversion of a real image taken through a camera on the ceiling. The quadtree integrates both obstacle regions and other regions in the workspace with its hierarchical structure in positioning. By using this hierarchical structure, the mobile robot is reduced to a point and then the forbidden regions where the robot cannot enter into are also understood in the quadtree. Hence, the algorithm can select the shortest collision-free path from the quadtree, i.e. a line between two given positions. Experimental results show that the proposed algorithm is superior to certain conventional algorithms with respect to calculation time. >

Bounds of eigenvalues of a graph

Abstract: LetG be a simple graph withn vertices. We denote by λi(G) thei-th largest eigenvalue ofG. In this paper, several results are presented concerning bounds on the eigenvalues ofG. In particular, it is shown that −1⩽λ2(G)⩽(n−2)/2, and the left hand equality holds if and only ifG is a complete graph with at least two vertices; the right hand equality holds if and only ifn is even andG℞2K n/2.

On short noncontractible cycles in embedded graphs

Abstract: This paper contains a proof that every triangulation of an orientable surface of genus $g > 1$ with n vertices contains a noncontractible cycle of length $O( \sqrt{n/g} \log g )$ These bounds are tighter than those previously known, but are not the conjectured optimal These results and techniques are related to “planarizing” problems in which a small set of vertices is sought whose removal leaves a planar graph and to “separator” problems in which a small set of vertices is sought whose removal leaves a graph with all components small

Search algorithm for Ramsey graphs by union of group orbits

Abstract: An algorithm for the construction of Ramsey graphs with a given automorphism group G is presented. To find a graph on n vertices with no clique of k vertices, Kk, and no independent set of l vertices, ¯Kl, k, l ≤ n, with an automorphism group G, a Boolean formula α based on the G-orbits of k-subsets and l-subsets of vertices is constructed from incidence matrices belonging to G. This Boolean formula is satisfiable if and only if the desired graph exists, and each satisfying assignment to α specifies a set of orbits of pairs of vertices whose union gives the edges of such a graph. Finding these assignments is basically equivalent to the conversion of α from CNF to DNF (conjunctive to disjunctive normal form). Though the latter problem is NP-hard, we present an “efficient” method to do the conversion for the formulas that appear in this particular problem. When G is taken to be the dihedral group Dn for n ≤ 101, this method matches all of the previously known cyclic Ramsey graphs, as reported by F. R. K. Chung and C. M. Grinstead [“A Survey of Bounds for Classical Ramsey Numbers,” Journal of Graph Theory, 7 (1983), 25–38], in dramatically smaller computer time when compared to the time required by an exhaustive search. Five new lower bounds for the classical Ramsey numbers are established: R(4, 7) ⩾ 47, R(4, 8) ⩾ 52, R(4, 9) ⩾ 69, R(5,7) ⩾ 76, and R(5, 8) ⩾ 94. Also, some previously known cyclic graphs are shown to be unique up to isomorphism.

Exotic n-universal graphs

Abstract: An n-universal graph is a graph that contains as an induced subgraph a copy of every graph on n vertices. It is shown that for each positive integer n > 1 there exists an n-universal graph G on 4n - 1 vertices such that G is a (v, k, λ)-graph, and both G and its complement G¯ are 1-transitive in the sense of W. T. Tutte and are of diameter 2. The automorphism group of G is a transitive rank 3 permutation group, i.e., it acts transitively on (1) the vertices of G, (2) the ordered pairs uv of adjacent vertices of G, and (3) the ordered pairs xy of nonadjacent vertices of G.

Method for tying and untying path access in a CPU-based, layered communications system

Abstract: A method for maintaining the integrity of ties and their associated tie groups in a CPU-based, layered communications subsystem in which the connection endpoints in each layer are denoted by a connection control block (CCB), the relationship between CCBs being denoted by ties. Ties and their CCBs can be mapped onto an edge-oriented graph of tie group relations. The arbitrary removal of an edge (tie) in the graph compromises graph integrity by possible formation of unenumerated subgraphs or independent graphs. The solution involves enumerating those edges having vertices which no longer reference CCBs within the tie group, removing them, and forming a second tie group. The enumeration is conducted over a Eulerian traverse of the remaining n edges of the graph. A Eulerian traverse of a graph is one which traverses each edge exactly once. Such a traverse reduces the number of comparisons M to a range N

Graph edge colorings and their chemical applications

Abstract: A general method is outlined to enumerate the edge-colorings of graphs under group action. The symmetry group of the graph acting on the vertices induces permutation of the edges. The edge-colorings are enumerated using the edge-permutation group. A number of chemical applications especially to multiple quantum NMR spectroscopy, statistical mechanics, enumeration of unsaturated isomers, etc. are considered.

Minimum length linear transistor arrays in MOS

Abstract: The following abstract problem is solved: given a two-terminal series-parallel (sp) graph, find an sp-equivalent graph in which the number of vertices with odd degree is minimum, where an sp-equivalence class consists of the graphs with the same series and parallel connections except for their order. The motivation for this problem is to find a minimum-length one-dimensional transistor array for MOS-discharge networks. >

Variants of the graph dependent model in extreme value theory

Abstract: With a set X1, X2, .... Xn n random variables, a graph is associated whose vertices are the integers 1,2,..., n and whose edges represent those pairs i and j for which the events {Xi>X} and {Xj>X} do not become “almost independent” for “large X”. With a variety of assumption on the edge set of the graph, the asymptotic distribution of the extremes of the Xj, when properly normalized, is determined. This refines the earlier result of the present author on this kind of dependence, and extends and unifies several known dependent extreme value models.

Hamiltonian properties of the cube of a 2‐edge connected graph

Abstract: Let G be a 2-edge connected graph with at least 5 vertices. For any given vertices a, b, c, and d in G with a ≠ b, there exists in G3 a hamiltonian path with endpoints a and b avoiding the edge cd, and there exists in G3 ∪ {cd} a hamiltonian path with endpoints a and b and containing the edge cd. Also, after removal of two edges or one edge and one vertex from G3, the resulting graph is still hamiltonian.

Longest ab-Paths in Regular Graphs

Abstract: Some results about longest paths between given vertices in regular graphs are given. One result is that if a graph G is regular of valence d, 3-connected and has at most 3d—1 vertices, then either G is a certain given graph, or G has a hamiltonian path joining any pair of vertices, except if G is bipartite and the pair of vertices belong to the same vertex class of G.

Algorithm and program

Abstract: In this paper, we briefly describe an algorithm and program for search for structural fragments which we have developed for the case when the compound is specified using the connectivity matrix. The considered problem may be formulated in the language of graph theory as follows. To each chemical structure is associated a graph with labeled vertices and edges. The vertices of the graph correspond to the atoms of the molecule; the edges correspond to the chemical bonds. The labels of the vertices and the edges code respectively the type of atom and bond. In this case, the search for structural fragments involves search for some specified subgraph (the structural fragment) in the graph (the original structure). This problem is well known in graph theory by the name of isomorphic embedding of graphs. There are several algorithms [I-ii] for the solution of the problem of isomorphic embedding of graphs [iIi]. Essentially, these algorithms are of the sorting type. In order to shorten the sorting process, we use specific necessary conditions for isomorphism, allowing us in a number of cases to narrow down the "forest" of the search. Let us outline the basic ideas of our proposed algorithm. In the graph of the original structure, we systematically select all the largest subgraphs with the number of vertices equal to the number of atoms in the structural fragment. For further investigation, we are interested only in those subgraphs in which the set of vertex labels coincides with the analogous set for the graph of the fragment. Then for each largest subgraph we decide the question of whether or not to eliminate some edges to obtain a subgraph isomorphic to the graph of the fragment. (For this purpose, a number of necessary conditions are tested.) If this is possible, then all such subgraphs are generated; then they are tested for isomorphism. The isomorphism test is a sorting process, and the sorting process is shortened on the basis of some necessary conditions for the isomorphism. The algorithm is distinguished from familiar procedures of the same name by the organization of the sorting process and the criteria for eliminating variantswhich are known to be unsuitable. The algorithm for search and retrieval of structural fragments has been realized in the form of a program in BASIC language on the Iskra-226 minicomputer. The input data for running the program are the adjacency matrices of the graphs of the original structure and the structural fragment. The program allows us to find all th~ subgraphs in the graph of the structure which are isomorphic to the graph of the fragment. The number of vertices (in increasing order) of the graph of the structure which form the desired subgraph and its adjacency matrix are displayed on the screen. The total number of subgraphs found are also indicated. A detailed description of the algorithm and the text of the program will be deposited.

A method of operating a digital computer to set routing paths

Abstract: A method of operating a digital computer to set routing paths along an x-y grid (10) of possible paths, between a plurality of terminals (60, 62, 64, 66, 68) on an integrated circuit, includes: a) determining an average position on the grid of the terminals (60,62, 64, 66, 68) to be interconnected; b) identifying a first terminal (64) closest to the average position along x and/or y paths of the grid; c) identifying a second terminal (62) closest to the first terminal (64) along x and/or y paths of the grid; d) establishing a first path (70) along x and/or y paths of the grid between the first and second terminals (64, 62), wherein the first path (70) is rectangular and has four edges (72A-72D) and four vertices (74A-74D) including the first and second terminals (64, 62) when the first and second terminals are on different x and different y paths, and wherein the first path (70) is a line and has one edge and two vertices including the first and second terminals when the first and second terminals are on the same x or same y path; e) setting the edge of the first path (70) as a routing path if the first path is a line; f) identifying a closest remaining terminal (66) which is closest along x and/or y paths of the grid, to an edge (72B) or vertex of an established path (70) having no set edges, or to a set edge or its vertices of an established path which has a set edge, whichever is closest; g) establishing a shortest path (76) along x and/or y paths of the grid between the closest terminal (66) and the edge or vertex of the established path to which it was closest, wherein the shortest path is rectangular and has four edges (78) and four vertices (80A, 80B) including the closest terminal (66) and a vertex common with a vertex of the established path (70) when the shortest path traverses both x and y paths of the grid, and wherein the shortest path is in line and has an edge and two vertices including the closest terminal when the shortest path traverses only x or only y paths of the grid; h) setting as a routing path edges (72A, 72B) of the established path (70) which couple its vertices (74A, 74C) which include terminals (62, 64) and which have a vertex in common with the shortest path, if the established path has no set edges and the common vertex does not include a terminal; i) setting as a routing path the edge of the shortest path (76) which couples the closest terminal to the edge (72B) or vertex of the established path (70) if the shortest path is a line and has a vertex common with a vertex of the established path, or is a line and has a vertex at a set edge of the established path; j) repeating steps f) to i) for remaining terminals to be interconnected.