Graph Algorithms · Hard problem

A graph has vertices $s,t,u,v,w,x,y,z$ and edges $st, su, tv, uv, uw, vx, wx, xy, wy, yz, xz$. Running a DFS from $s$ that visits each vertex's unvisited neighbors in alphabetical order discovers several back edges. What is the length (number of edges) of the LONGEST cycle that any single one of these back edges creates with its DFS tree path?