(I asked this on mathse. May be it suits better here)
How to find the cyclic ordering of edges incident on a vertex in a plane graph? (ie, in a plane embedding) Of coure we are looking for a polynomial time algorithm.
The algorithm i settled on uses dual graph. The idea was to take the set $E$ of edges incident on $v$. Take the set $F$ of faces that contains vertex $v$. Let the dual graph be $G^*$. Take the induced subgraph $G^*[F]$. It will be a cylcle and hence we have a cyclic ordering. From this, we can obtain a cylcic ordering of the edges incident on $v$ in the original graph.
In "Planar Formulae and Their Uses", Lichtenstein says
There is a choice of cyclic orderings of (neighbors of $v$ aruond $v$) ... for which NC(B) is planar if and only if G(B) is planar. Since any (polynomial) planarity algorithm can find such an ordering if it exists, ....
I understand that it is common not to be concerned with specific details of simle things in such papers.
But it seems i am missing something. Is there anything easier? Sorry; i know this must be a simple thing. But i don't see any better way.