Graph Algorithms · Hard problem

A network of $7$ computers numbered $1$–$7$ has connections $12, 13, 24, 25, 34, 36, 46, 57, 67$. A depth-first search starting at computer $1$, always visiting the lowest-numbered unvisited neighbor first, builds a search tree and finds several connections that are NOT part of that tree (back edges). Each back edge closes exactly one cycle with the tree path between its two endpoints. What is the length, in edges, of the shortest such cycle?