Figure 3

The four cases for introducing an edge uv with d ( u ) = d ( v ) = 2 in the dynamic programming for X ( i ): (a) there is a u - v -path in $G_j^S$, (b) u and v have dangling paths in $G_j^S$, (c) there is a u - x -path and a dangling path on v in $G_j^S$, or (d) there is a u - x -path and a v - y -path in $G_j^S$.