I have a list of interconnected edges (E), how to find the shortest path connecting from one vertex to another?
I am thinking about using lowest common ancestors, but the edges don't have a clearly defined root, so I don't think the solution works.
Shortest path is defined by the minimum number of vertexes treversed.
Note: There could be a multi-path connecting two vertices, so obviously breadth first search won't work
I'm not sure if you need a path between every pair of nodes or between two particular nodes. Since someone has already given an answer addressing the former, I will address the latter.
If you don't have any prior knowledge about the graph (if you do, you can use a heuristic-based search such as A*) then you should use a breadth-first search.
The Floyd-Warshall algorithm would be a possible solution to your problem, but there are also other solutions to solve the all-pairs shortest path problem.
Additional 2 cents. Take a look at networkx. There are interesting algos already implemented for what you need, and you can choose the best suited.
Shortest path is defined by the minimum number of vertexes treversed
it is same as minimum number of edges plus one.
you can use standard breadth first search and it will work fine. If you have more than one path connecting two vertices just save one of them it will not affect anything, because weight of every edge is 1.