The question asks for graphs that may suffice being the intersection of A{Bipartite Graph} & B{Complement of Path}
The way I understand this question is to find a complement of the bipartite graph that constitutes a path.
I know that every bipartite graph has paths in it but what about the complement of paths? Since Bipartite doesn't have to be complete and connected, I believe every bipartite can have it's complement being a path. So the answer is all bipartite suffice this condition.
I'm new to Graph Theory, so I'm not so sure if my logic here is sound.Thanks for any help!
If you're asking "Which bipartite graphs are the complements of paths?" then the answer is certainly not "All bipartite graphs." For example, $K_{3,3}$ is a bipartite graph:
but it's not the complement of a path, because its complement is a graph with two cycles:
In general, if either side of the bipartition is too large, the complement of the graph will contain a cycle. So you only have to check bipartite graphs in which both sides of the bipartition are small, and there will only be a few cases that work. Your answer will be a finite (and small) set of graphs.
(And a more promising direction to go in is actually posing the problem differently: "Which complements of paths are bipartite?" There are a lot fewer different graphs that are complements of paths to check, than there are bipartite graphs.)