In Cauchy theorem and also in some Alexandrov theorems I came across the words "If $P,P'$ are Combinatorially Equivalent Polyhedra",i.e those that have "same structure". Does that mean the following?:
There is a 1-1 and onto correspondece $f:V\to V'$, where $V, V'$ are the vertices of $P,P'$ and $v_1v_2...v_n$ is a face of $P$, if and only if $f(v_1)f(v_2)...f(v_n)$ is a face of $P'$ .
Why just isomorpic graphs of $P,P'$ is not enough here for the polyhedra to have the same structure?
Thanks.