In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a proper interval graph G and a particular set T of k vertices, the goal is to compute a path cover of G with minimum cardinality, such that the vertices of T are endpoints of these paths. We propose an optimal algorithm for this problem with ru...

An abstraction of the problem of multi-robot path planning is introduced in this paper. The basic task is to determine spatial-temporal plan for each robot of a group of robots where each robot is given its initial position in the environment and it needs to go to the given goal position. Robots must avoid obstacles and must not collide with each other. The abstraction adopted in this work mode...

The majority of the definitions used here will be found in (3),1 with the terms "point", "line", and "cycle" replaced by vertex, edge, and circuit. In particular, a block is a connected graph with no cut vertex, and a block is minimal if no spanning subgraph of the block with fewer edges is also a block. We consider the graph with one vertex and no edges (the vertex graph) and the graph with tw...

In a properly vertex-colored graphG, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P . If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring ofG. The minimum numbe...

Shortest Paths: Today we consider the problem of computing shortest paths in a directed graph. We are given a digraph G = (V,E) and a source vertex s ∈ V , and we want to compute the shortest path from s to every other vertex in G. This is called the single source shortest path problem. The algorithms we will present work for undirected graphs as well, by simply assuming that each undirected ed...

Path in a graph. The shortest path weight for a pair (u, v) of edges is defined as: δ(u, v) = ∞, if there is no path from u to v; δ(u, v) = min(w(p)): p connects u to v. A shortest path between u and v is any path p between them, such that w(p) = δ(u, v). Single-source shortest path problem. Given a graph G and a vertex v ∈ V , find the shortest path from v to every other node in the graph. Sin...