2-cell embeddings of graphs in surfaces and Euler formula

Question

I have a few questions regarding 2-cell embeddings of graphs in surfaces. Suppose $G$ is a 2-cell embedded graph in an orientable surface $S$,

a) Is any connected subgraph of $G$ 2-cell embedded in $S$?

b) Is the dual graph of $G$ 2-cell embedded in $S$?

I suspect 1) is false and 2) is true.

c) If 1) is false, under which circumstances will it be true?

Suppose now $G$ is embedded in an orientable surface $S$, but it may have non 2-cell faces, then, if $f$ gives the number of 2-cell faces, Euler formula gives that

$|V(G)|-|E(G)|+f\ge \chi(S)=2-2g$,

where $\chi(S)$ is the Euler characteristic and $g$ the genus (number of handles) of the surface.

d) Do you know of any reference for this?

Any help is greatly appreciated.

Thanks a lot in advance, and regards,

William.

Answer 1

The proof in C. Thomassen, Kuratowski’s Theorem, J. Graph Theory 5 (1981) 225-241, of Theorem 3.2 includes $v-e+f \geq \chi$.