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2-amalgamation 2-cell embedding 4-vertex graphs addition bipartite graph Bk+i and H cartesian products cB(G CmxCm complete graphs connected graph copy of vertex cross product crosscap defined distinct vertices edge e=uv edges of G embedding of G Euler characteristic Euler formula Example external face face containing G is connected girth g>3 graph G graph with genus Harary identifying induction inequalities isomorphic Lemma Let G minimal splitting minimum number n-1 splittings n-cube non-adjacent non-orientable genus number of CmxCn number of splittings number of vertices obtained from G orientable surface orientably simple Parameters of Graphs path of length planar embedding planar graph planar splitting plane products of 4-vertex proof re-identify resulting graph Ringel rungs shown in figure sphere split vertices splitting G splitting number splitting of G Stahl and Beineke subdivision subgraph homeomorphic subgraph of G third vertex Topological graph theory topological parameters torus unique embedding shown vertex identification vertex set