The Shortest Path Problem for a Multiple Graph

Abstract

In the article, the definition of an undirected multiple graph of any natural multiplicity $k > 1$ is stated. There are edges of three types: ordinary edges, multiple edges, and multi-edges. Each edge of the last two types is the union of $k$ linked edges, which connect 2 or $k + 1$ vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common ending vertex to $k$ linked edges of some multi-edge. If a vertex is the common end of a multi-edge, it cannot be the common end of any other multi-edge. Also, a class of the divisible multiple graphs is considered. The main peculiarity of them is a possibility to divide the graph into $k$ parts, which are adjusted on the linked edges and which have no common ordinary edges. Each part is an ordinary graph. The following terms are generalized: the degree of a vertex, connectedness of a graph, the path, the cycle, the weight of an edge, and the path length. The definition of a reachability set for the ordinary and multiple edges is stated. The adjacency property is defined for a pair of reachability sets. It is shown, that we can check the connectedness of a multiple graph with the polynomial algorithm based on the search for reachability sets and testing their adjacency. A criterion of the existence of a multiple path between two given vertices is considered. The shortest multiple path problem is stated. Then, we suggest an algorithm for finding the shortest path in a multiple graph. It uses Dijkstra’s algorithm for finding the shortest paths in subgraphs, which correspond to different reachability sets.