Bipartite Graphs and Matchings · Challenge problem

A network of $15$ devices splits into four pieces with no links between pieces: a $4$-cycle on $4$ devices, a $6$-cycle on $6$ devices, a path of $3$ devices (so $2$ links), and a single linked pair. In how many different ways can every device be labeled $A$ or $B$ so that every link joins an $A$-device to a $B$-device?