The Shortest Path Problem for a Multiple Graph

Аннотация

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.