Graph Algorithms · Challenge problem

A communications system has $12$ terminals split into two separate groups with NO connections between the groups, and $15$ connections in total: within the first group (terminals $1$–$7$): $(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(1,7),(2,4),(3,6)$; within the second (terminals $8$–$12$): $(8,9),(9,10),(10,11),(11,12),(8,12),(9,11)$. A depth-first search is run separately inside each group, starting from its lowest-numbered terminal. Across both groups combined, how many of the $15$ connections end up as back edges (not used by either search tree)?