Interesting Graph Theory "WOMVIES" problem Here is an interesting problem:
A graph is a set of vertices (points), some pairs of which are joined by an edge. For this problem, we will not allow an edge to join a
vertex to itself (i.e., no loops), nor will we consider pairs of vertices joined by more than one edge.
A subset S of vertices in a graph is called a womvie, (with one more vertex isolation ends), if
(1) no pair of vertices in S are joined by an edge, and
(2) the inclusion of any additional vertex in S destroys property (1).
For a graph G, M(G) will be used for the number of G's womvies.

M(C3) = 3,
M(C4) = _____, What are the womvies?
M(C5) = 5 , What are the womvies??
M(C6) = _____ C6 has 2 womvie sizes.
M(Cn) = 7 what is n?
{1,3,5} is a womvie in which figures?
So Im not sure how to approach this problem. Its interesting, but i dont really understand the definition of a "womvie" Anything to help? The problem is really confusing with all the parts, but please edit to make it less confusing.
Thank you
 A: A womvie is a set of vertices such that there is no edges between any of this vertex.
For cycles that mean that if a vertex $i$ belongs to the womvie then neither $i-1$ nor $i+1$ belong to the womvie.
Moreover you have an additional condition for the womvie that say somehow that they are maximal. In other word you cannot add any vertex in a womvie and get a set that is a womvie.
For cycles that means that if a vertex $i$ does not belong to the womvie than either $i-1$ or $i+1$ belong to the womvie.
For $C_3$, if you take a set with more than 1 vertices you are sure that two of them are connected by an edge. Hence all the possible womvie are composed of at most 1 vertex. There are 3 different vertices hence 3 womvie
For $C_4$, for a set of one vertex $i$ you can add the vertex $i+2$ (the other side of the diagonal) and be sure that the vertex are not connected. Hence there is no womvie that have size 1 in $C_4$. If you take a set of 3 or more vertices you are sure to have two vertices connected by an edge. Hence there is no womvie that have size 3 or more in $C_4$. The only possibility left are set of size 2. Only the sets composed of the end of the diagonals are not connected by an edge hence $M(C_4)=2$ and the womvie are $\{1,3\}$ and $\{2,4\}$.
I hope this clarify the definition of womvie and help you solve your problem.
ps:you can go through the other cases just as I did.
