NP-Completeness of the Eulerian Walk Problem for a Multiple Graph

In this article, we consider undirected multiple graphs of any natural multiplicity k > 1. A multiple graph contains edges of three types: ordinary edges, multiple edges, and multiedges. Each edge of the last two types is the union of linked edges that connect 2 or (k + 1) vertices, correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, then it can be incident to other multiple edges, and it can also be the common end of k linked edges of a multiedge. If a vertex is the common end of a multiedge, then it cannot be the common end of another multiedge. We study the problem of the Eulerian walk (cycle or trail) in a multiple graph, which generalizes the classical problem for an ordinary graph. We prove that the recognition variant of the multiple Eulerian walk problem is NP-complete. To do this, we first prove NP-completeness of the auxiliary problem of covering trails with the given endpoints in an ordinary graph.