Let G = (V,E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be t-approximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data structure for a given graph G which is capable of answering the following query for any u, v, z ∈ V and t > 1. rep...

Our rigorous path integral is extended to quantum evolution on metricaffine manifolds. 1 Path Integrals on Euclidean Spaces. Consider the evolution operator U [ψ(t, q)] = ψ(t, q), t ≥ t, of the quantum evolution equation on L(R): h̄ i ∂ψ(t, q)/∂t+ f(t, q, h̄ i ∂/∂q)ψ(t, q) = 0. Here the operator f(t, q, h̄ i ∂/∂q) is the standard quantization of a classical timedependent quasi-Hamiltonian f(t, q, ...

Let G = (V, E) be any undirected graph on V vertices and E edges. A path P between any two vertices u, v ∈ V is said to be t-approximate shortest path if its length is at most t times the length of the shortest path between u and v. We consider the problem of building a compact data structure for a given graph G which is capable of answering the following query for any u, v, z ∈ V and t > 1. re...

In the theory of graphs, there exist several algorithms that will find the shortest path between two vertices. Two of these algorithms are due to Dijkstra and Ford. However, an even more general problem would be to develop an algorithm that finds all shortest paths between two vertices in a graph. Furthermore, we may also want to look at minimal ways to cut the edges in a graph such that all sh...

We consider the following natural “above guarantee” parameterization of the classical Longest Path problem: For given vertices s and t of a graph G, and an integer k, the problem Longest Detour asks for an (s, t)-path in G that is at least k longer than a shortest (s, t)-path. Using insights into structural graph theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on undirec...

A tree T is a graph which is connected and doesn’t contain any circuits. Given any two vertices α ̸ = β ∈ T, let αβ be the unique path connecting α and β. Define the graph distance d(α, β) to be the number of edges contained in the path αβ. Let T be an infinite tree with root 0.The set of all vertices with distance n from the root is called the nth generation of T, which is denoted by L n . We d...

Given a graph G, and two vertex sets S and T of size k each, a many-tomany k-disjoint path cover of G joining S and T is a collection of k disjoint paths between S and T that cover every vertex of G. It is classified as paired if each vertex of S must be joined to a designated vertex of T , or unpaired if there is no such constraint. In this article, we first present a necessary and sufficient ...

For a connected graph G not isomorphic to a path, a cycle or a K1,3, let pc(G) denote the smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and t , respectively, then ...

Let C m T denote the Kronecker product of a cycle C m and a tree T. If m is odd, then C m T is connected, otherwise this graph consists of two isomorphic components. This paper presents a scheme which constructs a long cycle in each component of C m T, where T satisses certain degree constraints. The cycle thus traced is shown to be a dominating set, and in some cases, a vertex cover of that co...

A model of visual apparent motion is derived from four observations on path selection in ambiguous displays in which apparent motion of illuminated dots could, in principle, be perceived along many possible paths: (a) Whereas motion over each path is clearly visible when its stimulus is presented in isolation, motion is usually seen over only one path when two or more such stimuli are combined ...