The interior of a polygon with n sides and decomposed into 7 triangles and 4 squares was found to be a planar representation of a graph with 11 vertices. Determine n
I tried drawing various figures with that number of triangles and squares and use the Euler’s formula. But I cannot see any relation between n and the vertices/faces/edges. Any hints to start pls?
We are given $V = 11$ vertices in total. The number of faces is clearly $F = 7 + 4 + 1 = 12$, since $7$ faces are triangular, $4$ are square, and there is one more face for the region outside of the polygon.
All that is left is to count the number of edges. There are $3$ edges in each triangle, $4$ in each square, and $n$ edges from the original polygon. But each edge is shared by exactly two faces, so the total number of edges is $$E = \frac{3(7) + 4(4) + n(1)}{2} = \frac{n+37}{2}.$$ With all of this, now can you apply Euler's formula? Finally, explicitly construct such a graph to demonstrate that it is in fact possible to meet the criteria of the problem.