Showing papers on "Path graph published in 1974"
TL;DR: In this article, the state space is the set of vertices of a connected graph, and for each vertex the transition is always to an adjacent vertex, such that each of the adjacent vertices has the same probability.
248 citations
TL;DR: A sufficient condition is given, based on the minimumdegree of the vertices of G and the maximum degree of the Vertices of H, for H to be a subgraph of G.
61 citations
TL;DR: In this article, the least integer N such that for every graph G with n vertices, either G contains a path of m vertices or the complement of G contain a vertex of degree at least n.
41 citations
TL;DR: It is proved that if a 4-critical graph I' has a vortex x of large valency (compared to the number of vertices of I' not adjacent to x), then I' contains Vertices of valency 3, and a list of all 4- critical graphs with @? 9 vertices is exhibited.
39 citations
TL;DR: These algorithms generalize the Floyd algorithm and the Dantzig algorithm for finding the shortest path between every pair of vertices in a directed graph.
Abstract: Two algorithms are presented that construct the k shortest paths between every pair of vertices in a directed graph. These algorithms generalize the Floyd algorithm and the Dantzig algorithm for finding the shortest path between every pair of vertices in a directed graph.
36 citations
TL;DR: In this paper, the authors present a few methods of deter-mining the minimum possible width in some cases that arise in the numerical solution of Laplace's equation for the common square grid and the cubic grid.
Abstract: The incidence matrices of various graphs are considered. By reordering the points, the bandwidth can be changed. In the cases of rectangular grids in the plane or cubic grids in three dimensions, the exact, minimum values of the bandwidth are deter- mined. In certain numerical analysis problems, it is of interest to index the vertices of a graph in such a way that the matrix used to represent an associated system of linear equations is as close to diagonal as possible or, equivalently, to index the vertices of a graph in a way that minimizes the width of the band of nonzero terms in the incidence matrix for that graph. The purpose of this note is to present a few methods of deter- mining the minimum possible width in some cases that arise in the numerical solution of Laplace's equation. Of particular interest are the first proof of Theorem 2 which solves the problem for the common square grid and Theorem 3 which answers the prob- lem for the cubic grid.
20 citations
TL;DR: In this article, the minimum number of edges and vertices in a graph with edge connectivity n and exactly m n-bonds cuts was studied, and the problem was shown to be NP-hard.
Abstract: The problem studied is the following: What is the minimum number of edges and vertices in a graph with edge connectivity n and exactly m n-bonds cuts? It is perhaps surprising that this problem tur...
19 citations
TL;DR: In this article, the maximum number of edges of a graph Gn of n vertices if Gn does not contain I12 as a subgraph was shown. But the problem of finding the subgraph that contains I12 was not addressed.
18 citations
TL;DR: This paper sharpens Manvel's result by showing that if the graph is not a triangulation of a hexagon, then reconstruction can be accomplished using only those isomorphism types of subgraphs corresponding to deletion of vertices of valence two.
3 citations
1 citations