Covering projections of graphs preserving links of vertices and edges (Q1339869)

The combinatorial covering projection is defined; it is an analogy of covering projections of topological spaces. If \(G\) and \(H\) are graphs, then the combinatorial covering projection is a morphism \(f: H\to G\) for which there exist orientations of \(G\) and \(H\) such that \(f\) preserves directions of edges and for each vertex \(u\) of \(G\) the morphism \(f\) maps the set of edges originated at \(u\) in \(G\) in a one-to-one way onto the set of edges originating at \(f(u)\) and maps the set of edges terminating at \(u\) in a one-to-one way onto the set of edges terminating at \(f(u)\). Further the links in graphs are defined. The symbol \(\text{Link}_ 0(u,G)\) denotes the subgraph of a graph \(G\) formed by the vertex \(u\) and all the edges of \(G\) incident to \(u\). The symbol \(\text{Link}_ m(u,G)\) for a positive integer \(m\) denotes the subgraph of \(G\) induced by the set of all vertices whose distance from \(u\) in \(G\) is less than or equal to \(m\). Some theorems concerning these concepts are proved.