Surface expansions of graphs (Q2767623)

Let \(G\) be a finite connected graph, let \(P\) be a compact surface with border, and let \(i:G\to P\) be an embedding. The pair \((P,i)\) is called surface expansion of the graph \(G\) if the image \(i(G)\) is a deformation retract of \(P\). Two expansions \((P_1,i_1)\) and \((P_2,i_2)\) of the same graph \(G\) are equivalent if the surface \(P_1\) is homeomorphic to \(P_2\). The author proves that the graph \(G\) has \(-\chi+1-2g\) nonequivalent nonoriented surface expansions and the number of nonequivalent oriented surface expansions does not exceed \([-\chi/2-g]+2\), where \(\chi\) is the Euler characteristic and \(g\) is the maximal oriented genus of \(G\).