Your task is to study one or more of the following three classes of graphs:

• Class A consists of those graphs G in which every pair of distinct vertices belongs to precisely one common 4-cycle in G. That means, for every x!=y in V(G), there always is a 4-cycle in G passing through both x,y, but only one such. For example, the graph C₄ belongs to A.
• Class B consists of those graphs G in which every pair of distinct non-adjacent vertices belongs to precisely one common 4-cycle in G. That means, for every x!=y in V(G) such that xy is not an edge, there always is a 4-cycle in G passing through both x,y, but only one such. For example, the graphs C₄ and K₄ (and even larger complete graphs) belong to B.
• Class C consists of those graphs G in which every pair of distinct vertices belongs in G to precisely one 4-cycle in which these two vertices are not consecutive. That means, for every x!=y in V(G), there always is a 4-cycle in G passing through both x,y and not using the (possible) edge xy, but only one such. If xy happens to be an edge of G, then we do not care about the number of 4-cycles passing through the edge xy. For example, the graph K₄ belongs to C (but C₄ does not).

For start, choose your favourite class(es) among A,B,C and try to find more graphs which belong to them. How many can you find? How much can you say about your chosen class(es)? Can you find all the graphs in your class(es) and prove, that there are no more graphs there?

These questions are rather difficult, and the assignment is intentionally left "open". Try to solve as much as you can, and you will receive bonus points based on how your solution compares to other submitted solutions (of course, the better/richer your solution is than the other solutions, the more points you would get).