Let all vertices be numbered from 1 to n at clockwise order starting from some vertice. Let's make such sequence b of n elements: if i-th vertice is connected with j and i < j, then b[i] = '(', else b[i] = ')'. Easy to understand, that if graph was correct, than b — is correct bracket sequence, and if graph wasn't correct than b is not correct bracket sequence. So, problem is equivalent to counting number of correct bracket sequences of length 2n and well-known that answer is n-th Catalan number.