In a graph G with non-negative edge lengths, let P be a shortest path from a vertex s to a vertex t. We consider the problem of computing, for each edge e on P , the length of a shortest path in G from s to t that avoids e. This is known as the replacement paths problem. We give a linearspace algorithm with O(n log n) running time for n-vertex planar directed graphs. The previous best time boun...

We consider regular tessellations of the plane as infinite graphs in which q edges and q faces meet at each vertex, and in which p edges and p vertices surround each face. For 1/p + 1/q = 1/2, these are tilings of the Euclidean plane; for 1/p + 1/q < 1/2, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distanc...

The optimal path cover problem is to find a minimum number of vertex disjoint paths which together cover all the vertices of the graph. Finding an optimal path cover for an arbitrary graph is known to be NP-complete [3]. However, polynornial-time algorithms exist for trees [4], cacti [4] and for interval graphs [9]. The solution presented in [2] For circular-arc graphs is known to be wrong. In ...

The vertex set and arc set of a digraph D are denoted by V (D) and E (D), respectively, and the number of vertices in a digraph D is denoted by n (D). A directed cycle (path, walk) in a digraph will simply be called a cycle (path, walk). A graph or digraph is called hamiltonian if it contains a cycle that visits every vertex, traceable if it contains a path that visits every vertex, and walkabl...

A vertex-coloured graph G is said to be rainbow vertex-connected, if every two vertices of are connected by a path whose internal have distinct colours. The vertex-connection number G, denoted rvc(G), the smallest colours that needed make vertex-connected. This study focuses on deriving formulas for vertex connectivity simple ladder and roach graph.

In the previous chapter, we saw algorithms to find the shortest path from a source vertex s to a target vertex t in a directed graph. As it turns out, the best algorithms for this problem actually find the shortest path from s to every possible target (or from every possible source to t) by constructing a shortest path tree. The shortest path tree specifies two pieces of information for each no...

An algorithm is presented for computing geodesic furthest neighbors for all the vertices of a simple polygon, where geodesic denotes the fact that distance between two points of the polygon is defined as the length of an Euclidean shortest path connecting them within the polygon. The algorithm runs in O(n log n) time and uses O(n) space; n being the number of vertices of the polygon. As a corol...