Given a directed graph with the following attributes: - a chain from node $a$ to node $z$ passing nodes $b$ to $y$ exists and is unidirectional. - additionally a set of nodes having bidirectional vertices to at least two of the nodes $a \ldots z$ exists. These nodes are connected in a second unidirectional chain.
(the red route is the requested result, the squares are the first chain (unidirectional) and the circles are the second chain (unidirectional). $1$ is the start node and $5$ is the destination node.)
Is it possible to find the shortest path from $a$ to $z$ that includes nodes $b$ to $y$ and the additional nodes once without probing all possibilities?
I think the problem is roughly the same as the minimal traveling salesman problem since adding a vertex from $z$ to $a$ will result in the min-TSP - but this problem is slightly easier since a path from $a$ to $z$ is already known.