Minimum length walk from s to t covering a subset of vertices
So you are given a weighted undirected graph $G$. You want to find a shortest walk from $s$ to $t$ such that you visit all the vertices of the set $S$. You may choose to visit vertices in $V \setminus S$. You may visit vertices multiple times. You may use the edges multiple times. Let's call this problem $stSWALK$.


  $stSWALK$ is equivalent to metric Traveling Salesman Path problem ($MTSPP$).


It is easy to see that, when $S = V$ then $stSWALK$ is same as $MTSPP$.

You can convert an instance of $stSWALK$ into an instance of $MTSPP$ by simply computing the shortest paths between every pair of vertices in $S$ (and adding the corresponding "new" edges with the weights equal to the lengths of these shortest paths) and then solving $MTSPP$ on the graph induced by $S \cup \{s,t\}$ (lets call this graph $H$). To get a solution to $stSWALK$, we simply expand these newly added edges into their corresponding shortest paths in $G$.

Note that If $G$ is planar then $H$ is not necessarily a planar graph.