2013 AMC 12B Problem 12

Cities $C$ and $E$ each touch only two roads, so they behave like detours on the $A$–$D$ and $B$–$D$ connections

Count the trails from $A$ to $B$ on the reduced graph, then multiply by the independent detour choices

City $E$ (roads $AE, DE$) is a detour on an $A$–$D$ trip, and city $C$ (roads $BC, CD$) is a detour on a $B$–$D$ trip. Replace them to get a graph on $A, B, D$ with two $A$–$D$ connections, two $B$–$D$ connections, and one $A$–$B$ road. The trails from $A$ to $B$ using each once are of $4$ types: $ABDADB,$ $ADABDB,$ $ADBADB,$ and $ADBDAB.$ Each detour (through $E,$ through $C$) can be taken on either passage, so each type gives $4$ actual routes, for $4\cdot 4 = 16$ routes. Thus, the correct answer is D.