  Given an undirected graph $G=(V,E)$ and vertices $s,t,w_1,w_2\in V$, we study
finding whether there exists a simple path $P$ from $s$ to $t$ such that
$w_1,w_2 \in P$. As a sub-problem, we study the question: given an undirected
graph and three of its edges, does there exist a simple cycle containing all
those edges? We provide necessary and sufficient conditions for the existence
of such paths and cycles, and develop efficient algorithms to solve this and
related problems.
