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how i determine a face of a planar graph is convex polygon or not..............

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I can't make sense of this as it stands. A planar graph is one which can be embedded into the plane. Your question only makes sense for an actual emebedding in the plane, and I cannot see any sensible answer save that a face is a convex polygon if it's a convex polygon. Perhaps a more precise question is lurking? – Robin Chapman Jun 15 2010 at 6:37
With a protractor, I suppose. You may want to expand on your question to make it easier to know what exactly you are asking. – Gerry Myerson Jun 15 2010 at 6:38
I'm going to take a wild guess and say the question is asking whether one can tell if a face of a planar graph is a convex polygon before the graph is drawn with edges as straight line segments. That is, in the beginning, you are given a random drawing where the edges could be curved. Still, I believe the question will be closed unless the OP makes it clear what is being asked. – Gjergji Zaimi Jun 15 2010 at 7:32
Perhaps you are asking for the result of Steinitz's Theorem: Which planar graphs are $1-$skeletons of convex polytopes. Answer: The underlying abstract graph has to be $3-$edge connected. Such graphs have essentially (up to isotopies and an orientation reversing homeomorphism) only one planar embedding coming from a realization as the $1-$skeleton of a convex polytope. – Roland Bacher Jun 15 2010 at 8:37
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 The question has been closed: we simply don't know what you're asking. If you edit it to provide enough information, precision and context so that we can understand it (and, for bonus points, use proper punctuation and grammar, to the best of your ability), it can be reopened. – Pete L. Clark Jun 15 2010 at 15:14
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closed as not a real question by Robin Chapman, Andrew Stacey, François G. Dorais♦, S. Carnahan♦, Pete L. Clark Jun 15 2010 at 15:12

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this is not a well defined question. Perhaps you mean that you want to know if a face of a planar embedding of a planar graph is convex ? Note that it's always possible to draw a planar graph so all faces (with the exception of the outer face) are convex

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 (I think that this should have been left as a comment, not an answer) – Andrew Stacey Jun 15 2010 at 7:54

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