On Existence of Must-Include Paths and Cycles in Undirected Graphs

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.