Showing papers on "Path graph published in 1981"
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: A random graph with (1+ε)n/2 edges contains a path of lengthcn.
Abstract: A random graph with (1+e)n/2 edges contains a path of lengthcn. A random directed graph with (1+e)n edges contains a directed path of lengthcn. This settles a conjecture of Erdos.
146 citations
TL;DR: A graph having 27 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.
Abstract: A graph having 27 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.
88 citations
TL;DR: An extension of a conjecture of Hobbs, a new proof of Tutte's theorem on 3-connected graphs, and a result on the existence of a vertex joined by edges to three vertices of a cycle in a graph are obtained.
84 citations
TL;DR: In this article, the authors define R(n) to be the maximal number of pairwise orthogonal one-factorizations of the complete graph onn vertices, where "orthogonal" means that any two one-factors involved have at most one edge in common.
Abstract: The existence of a Room square of order 2n is known to be equivalent to the existence of two orthogonal one-factorizations of the complete graph on 2n vertices, where “orthogonal” means “any two one-factors involved have at most one edge in common.” DefineR(n) to be the maximal number of pairwise orthogonal one-factorizations of the complete graph onn vertices.
40 citations
TL;DR: The problem of how ''near'' the authors can come to a n-coloring of a given graph is investigated and an @W(epn) deterministic algorithm for finding such an n-color assignment is exhibited.
40 citations
TL;DR: This chapter discusses the relative lengths of paths and cycles in k -connected graphs and proves that if a graph G contains a cycle of length l, then G also contains a path of length at least l – 1.
30 citations
TL;DR: This work constructs infinite families of graphs having identity automorphism group, yet every vertex is pseudosimilar to some other vertex, and constructs, for each n, graphs containing a subset of vertices of size n which are mutually Pseudosimilar.
Abstract: : Dissimilar vertices whose removal leaves isomorphic subgraphs are called pseudosimilar We construct infinite families of graphs having identity automorphism group, yet every vertex is pseudosimilar to some other vertex Potential impact on the Reconstruction Conjecture is considered We also construct, for each n, graphs containing a subset of vertices of size n which are mutually pseudosimilar The analogous problem for mutually pseudosimilar edges is introduced (Author)
19 citations
TL;DR: The set Fn of all pairs (a, b) of integers such that there is a graph G with n vertices and binding number a/b that has a realizing set of b vertices is characterized.
Abstract: The concept of the binding number of a graph was introduced by Woodall in 1973. in this paper we characterize the set Fn of all pairs (a, b) of integers such that there is a graph G with n vertices and binding number a/b that has a realizing set of b vertices.
13 citations
TL;DR: In this article, the authors characterized non-hamiltonian graphs with n vertices, which satisfy the Ore-type degree-conditiond(x)+d(y)≥n−2 for each pairx,y∉M of different nonadjacent vertices whereM consists of two vertices of G. As an application a theorem on hamiltonian connectivity of graphs is given.
Abstract: Those non-hamiltonian graphsG withn vertices are characterized, which satisfy the Ore-type degree-conditiond(x)+d(y)≥n−2 for each pairx,y∉M of different nonadjacent vertices whereM consists of two vertices ofG. As an application a theorem on hamiltonian connectivity of graphs is given. Furthermore, a condition is presented which is sufficient for the existence of a covering of a graph by two disjoint paths with prescribed set of startpoints and prescribed set of endpoints. A class of graphs is described which have no covering of this kind.
12 citations
TL;DR: It is shown that if three vertices of the graph G(l') of a convex 3-polytope P are chosen, then G(P) contains a refinement of the complete graph C"4 on four vertices, for which the three chosen vertices are principal.
01 Jan 1981
TL;DR: It is shown that every k-edge-connected digraph with m edges and n vertices contains a spanning connected subgraph having at most 2m + 6(k −1)(n − 1))(5k − 3) edges.
01 Jan 1981
01 Jan 1981
TL;DR: In this paper, a new proof is given that any m-graph with at least one edge of multiplicity m is connected, and the construction of this path also establishes best possible upper and lower bounds on the length of the shortest path between any two vertices of a given mgraph.
Abstract: An m-graph is a graph, without loops, but with multiple edges of any multiplicity less than or equal to m. An exact m-graph is an m-graph with at least one edge of multiplicity m. A new proof is given that the graph \(R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))\), of all m-graphic realisations of a degree sequence, \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d}\), is connected. This is done by taking any two vertices of \(R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))\), say G and H, and finding a path between them which preserves any previously chosen edge of multiplicity m that occurs in both G and H. The construction of this path also establishes best possible upper and lower bounds on the length of the shortest path between any two vertices of \(R(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{d} ,L(m))\).
TL;DR: The distribution of the edges in level 3 in a deficient 3-generalized Fibonacci graph is investigated, and tools that might be useful in extending the results to higher levels are developed.
TL;DR: A representation theorem is proved for the cofactors of a special class of matrices which contains the tree matrices associated with graphs which minimizes the number of spanning trees of the multigraph obtained from G by adding μ parallel edges between every pair of distinct vertices.
01 Jan 1981