Topic
Path graph
About: Path graph is a research topic. Over the lifetime, 2613 publications have been published within this topic receiving 53539 citations.
Papers published on a yearly basis
Papers
More filters
TL;DR: The main result of this paper is that if the authors can draw a graph on a surface of genus g, then they can bisect it by removing $O(\sqrt{gn})$ vertices, best possible to within a constant factor.
293 citations
TL;DR: A method to compute path expressions by dividing G into components, computing path expressions on the components by Gaussian elimination, and combining the solutions is described, which requires O(m $\alpha$(m,n) time on a reducible flow graph.
Abstract: Let G = (V,E) be a directed graph with a distinguished source vertex s. The single-source path expression problem is to find, for each vertex v, a regular expression P(s,v) which represents the set of all paths in G from s to v. A solution to this problem can be used to solve shortest path problems, solve sparse systems of linear equations, and carry out global flow analysis. We describe a method to compute path expressions by dividing G into components, computing path expressions on the components by Gaussian elimination, and combining the solutions. This method requires O(m $\alpha$(m,n)) time on a reducible flow graph, where n is the number of vertices in G, m is the number of edges in G, and $\alpha$ is a functional inverse of Ackermann''s function. The method makes use of an algorithm for evaluating functions defined on paths in trees. A simplified version of the algorithm, which runs in O(m log n) time on reducible flow graphs, is quite easy to implement and efficient in practice.
290 citations
TL;DR: This paper presents an efficient algorithm for all k>cn0.5(log n)0, for any fixed c>0, thus improving the trivial case k>, and based on the spectral properties of the graph.
Abstract: We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n, 1/2), and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kucera. In this paper we present an efficient algorithm for all k>cn0.5, for any fixed c>0, thus improving the trivial case k>cn0.5(log n)0.5. The algorithm is based on the spectral properties of the graph. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 457–466, 1998
285 citations
TL;DR: An efficient technique for on-line planarity testing of a graph is presented that uses O(n) space and supports tests and insertions of vertices and edges in O(\log n) time, where n is the current number of Vertices of G.
Abstract: The on-line planarity-testing problem consists of performing the following operations on a planar graph $G$: (i) testing if a new edge can be added to $G$ so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for on-line planarity testing of a graph is presented that uses $O(n)$ space and supports tests and insertions of vertices and edges in $O(\log n)$ time, where $n$ is the current number of vertices of $G$. The bounds for tests and vertex insertions are worst-case and the bound for edge insertions is amortized. We also present other applications of this technique to dynamic algorithms for planar graphs.
283 citations
TL;DR: It is shown that every 2-connected triangulated planar graph with n vertices has a simple cycle C of length at most 2√2 · n which separates the interior vertices A from the exterior vertices B such that neither A nor B contain more than 2 3n vertices.
260 citations