The Polynomial Algorithm of Finding the Shortest Path in a Divisible Multiple Graph

In this paper, we study undirected multiple graphs of any natural multiplicity к > 1. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of к linked edges, which connect 2 or (к + 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 end of к linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. Divisible multiple graphs are characterized by a possibility to divide the graph into к parts, which are adjusted on the linked edges and which have no common edges. Each part is an ordinary graph. As for an ordinary graph, we can define the integer function of the length of an edge for a multiple graph and set the problem of the shortest path joining two vertices. Any multiple path is a union of к ordinary paths, which are adjusted on the linked edges of all multiple and multi-edges. In the article, we show that the problem of the shortest path is polynomial for a divisible multiple graph. The corresponding polynomial algorithm is formulated. Also we suggest the modification of the algorithm for the case of an arbitrary multiple graph. This modification has an exponential complexity in the parameter к.