Showing papers on "Path graph published in 1992"
TL;DR: In this chapter, the structure of the graph is completely encoded, so that, given the labels of two vertices, one can test if they are adjacent in time linear in the size of the labels.
Abstract: How to represent a graph in memory is a fundamental data structuring question. In the usual representations of an n-vertex graph, the names of the vertices (i.e., integers from 1 to n) betray nothing about the graph itself. Indeed, the names (or labels) on the n vertices are just $\log n$ bit place holders to allow data on the edges to encode the structure of the graph. In this scenario, there is no such waste. By assigning $O(\log n)$ bit labels to the vertices, the structure of the graph is completely encoded, so that, given the labels of two vertices, one can test if they are adjacent in time linear in the size of the labels. Furthermore, given an arbitrary original labeling of the vertices, structure coding labels are found (as above) that are no more than a small constant factor larger than the original labels. These notions are intimately related to vertex-induced universal graphs of polynomial size. For example, planar graphs can be labeled with structure coding labels of size $ < 4\log n$, which i...
255 citations
TL;DR: With a slight extension and modification of the algorithm, the maximum weight k -independent set problem on an interval graph with n vertices can be solved in O ( n k ) time using O (n k ) space.
94 citations
TL;DR: This paper partially answer the question: how slowly must p(n) converge to 0 so that a random graph K(n, p) has property PMG almost surely, where PMG means that all n vertices can be covered by vertex-disjoint copies of a fixed graph G.
42 citations
Proceedings Article•
01 Sep 1992TL;DR: In this paper, it was shown that the performance ratio of any vertex-coloring algorithm can be no better than O(n/log 2 n), even for randomized algorithms against oblivious adversaries.
Abstract: An algorithm for vertex-coloring graphs is said to be online if each vertex is irrevocably assigned a color before any later vertices are considered. We show that such algorithms are inherently ineffective. The performance ratio of any such algorithm can be no better than O(n/log2 n), even for randomized algorithms against oblivious adversary.We also show that various means of relaxing the constraints of the on-line model do not reduce these lower bounds. The features include presenting the input in blocks of log2 n vertices, recoloring any fraction of the vertices, presorting vertices according to degree, and disclosing the adversary's previous coloring.
31 citations
TL;DR: It is shown how to test at any time whether two vertices belong to the same 2-edge-connected component, and how to insert and delete an edge in O(m^{2/3} ) time in the worst case, where m is the current number of edges in the graph.
Abstract: This paper studies the problem of maintaining the 2-edge-connected components of a graph undergoing repeated dynamic modifications, such as edge insertions and edge deletions. It is shown how to test at any time whether two vertices belong to the same 2-edge-connected component, and how to insert and delete an edge in $O(m^{2/3} )$ time in the worst case, where m is the current number of edges in the graph. This answers a question posed by Westbrook and Tarjan [Tech. Report CS-TR-229-89, Dept. of Computer Science, Princeton University, Princeton, NJ, August 1989; Algorithmica., to appear].For planar graphs, the paper presents algorithms that support all these operations in $O(\sqrt {n\log \log n} )$ worst-case time each, where n is the total number of vertices in the graph.
30 citations
TL;DR: This paper presents an O(n log n) algorithm for the problem with n vertices in the forest and uses the algorithm to schedule real-time jobs with minimum distance constraints.
28 citations
TL;DR: It is proved that there exists a function f(k, n) such that for any colored graph G, if χ(G)>f(ω(G), n) then G induces either a colorful out directed star with n leaves or a colorful directed path on n vertices.
26 citations
TL;DR: Hilton's theorem is improved in this paper that G contains at least ⌊k2⌋ edge-disjoint 1-factors.
16 citations
TL;DR: A new linear-time algorithm to construct a rectilinear planar layout (horvert-representation, visibility representation) for a given planar graph based on the canonical representation of planar graphs is presented.
15 citations
TL;DR: Dirac proved that if each vertex of a graph G of order n ⩾3 has degree at least n /2, then the graph is Hamiltonian, and this result will be generalized by showing that if the union of the neighborhoods of each pair of vertices of a 2-connectedgraph G of sufficiently large order n has at leastn /2 vertices, then G isHamiltonian.
14 citations
TL;DR: A simple linear time algorithm for finding a cycle through three fixed vertices in an undirected graph based on decompositions into triconnected components and on a combinatorial result of Lovasz is presented.
TL;DR: The minimum, taken over all graphs G, is found to be at least ⌊ n 2 /4⌋ + 1 edges, of the number of vertices and edges of G which are on cycles of length 2 k + 1.
TL;DR: The purpose of this note is to demonstrate that such algorithms can be made to accommodate negative-weight vertices without loss of efficiency.
TL;DR: In this paper, the minimum number of edges in a graph on n vertices with the property that after deleting any edge, the remaining graph has diameter no more than D was given.
TL;DR: It is shown that every weakly triangulated graph not containing as an induced subgraph a path on five vertices or the complement of a paths on six vertices is charming.
TL;DR: It is proved that, for every integer n ⩾ 3, f(n) ⩽ n + k + [( 1 2 ( 8n − 24k 2 + 8k − 7 − 1)] where k = [ 1 21 ( 21n − 26 + 11)] and upper and lower bounds on f2(n are obtained.
TL;DR: If G is a graph on n vertices with n ≥ k2(k – 1)/4 and m < t(n, k) edges, then G contains a complete subgraph Kk such that the sum of the degrees of the vertices is at least 2km/n.
Abstract: Let t(n, k) denote the Turan number—the maximum number of edges in a graph on n vertices that does not contain a complete graph Kk+1. It is shown that if G is a graph on n vertices with n ≥ k2(k – 1)/4 and m < t(n, k) edges, then G contains a complete subgraph Kk such that the sum of the degrees of the vertices is at least 2km/n. This result is sharp in an asymptotic sense in that the sum of the degrees of the vertices of Kk is not in general larger, and if the number of edges in G is at most t(n, k) – ϵ (for an appropriate ϵ), then the conclusion is not in general true. © 1992 John Wiley & Sons, Inc.
TL;DR: A new path planning method for obstacle avoidance of mobile robot, where the free space of the environment is represented by a skeleton consisting of wires and nodes, whereas the skeleton of free space is obtained based on the thinning algorithm which is widely used in the image processing technique.
TL;DR: G(p, d) as discussed by the authors is a cubic (3-valent) graph consisting of a p-gon and a (p/d)-gon with corresponding vertices joined (the notation admits anomalous cases, when d=1 or (d, p)>1).
Abstract: G(p, d) is a cubic (3-valent) graph consisting of a p-gon and a (p/d)-gon (a starpolygon) with corresponding vertices joined (the notation admits anomalous cases, when d=1 or (d, p)>1), and with a high degree of symmetry. It is shown here that the seven possible graphs G(p, d) are just the edge-graphs of the regular polyhedra of type {p, 3} with 2p vertices, and therefore 3p edges, 6 faces, and symmetry group of order 12p.
TL;DR: Finite undirected loopless graphs G in which multiple edges are possible are considered, and it is shown that P 1,..., P k, C 1 ,..., C l are pairwise edge-disjoint.
Abstract: We consider finite undirected loopless graphs G in which multiple edges are possible. For integers k, l ≥ 0 let g(k, l) be the minimal n ≥ 0 with the following property: If G is an n-edge-connected graph, s 1 ,..., s k , t 1 ,..., t k are vertices of G, and f 1 ,..., f l , g 1 ,..., g 1 are pairwise distinct edges of G, then for each i = 1,..., k there exists a path P i in G connecting s i and t i and for each i = 1,..., l there exists a cycle C i in G containing f i and g i such that P 1 ,..., P k , C 1 ,..., C l are pairwise edge-disjoint
01 Jan 1992
TL;DR: In this paper, it was shown that if T and T′ are two vertices of a simple tree graph such that d_T(G)(T)(?)d_T (G}(T′), then there are n-disjoint paths in T(G) joining them.
Abstract: Let T(G)be the tree graph of a simple graph G.It is proved that if T and T′are two vertices of T(G)such that d_T(G)(T)(?)d_T(G}(T′),then there ared_T(G)(T) internally disjoint paths in T(G) joining T and T′.
TL;DR: This work investigates the behavior of the function f = f(n,k,e) defined as the smallest integer with the following property: If in a graph on n vertices, the numbers of edges in any two induced subgraphs on k vertices differ by at most e, then the graph or its complement has at most f edges.
Abstract: We investigate the behavior of the function f = f(n,k,e) defined as the smallest integer with the following property: If in a graph on n vertices, the numbers of edges in any two induced subgraphs on k vertices differ by at most e, then the graph or its complement has at most f edges. One of the results states that f(n,k,e) = e for n ≥ (2√2 + e)e 3/2 and 2e + 2 ≤ k ≤ n - 4e - 3
01 Jan 1992
TL;DR: This paper presents a data structure to store materialized k-level transitive closure such that retrievals and updates of a k- level transitiveclosure with path information being kept may be performed efficiently.
Abstract: A k-level transitive closure of a directed graph is all pairs of vertices (x, y) such that there exists at least a path from x to y of length d, d≤k. Multiple edges between a pair of vertices are allowed in a graph. This paper presents a data structure to store materialized k-level transitive closure such that retrievals and updates of a k-level transitive closure with path information being kept may be performed efficiently.