Terminal Access in Complementary Spanning Trees with At Most Two Missing Neighbors

Let a loopless multigraph be the physically labelled union of two disjoint spanning trees A and B. We prove that if some vertex is nonadjacent to at most two other vertices, every physical edge of either tree can occur in the first pair of an ordering for which every contiguous exchange into B is a spanning tree. The mate is freely chosen, and no initial interval-interchangeable ordering is assumed. The arbitrary-rank argument uses rooted reductions, simultaneous path expansion, and two exhaustively qualified finite certificate domains. The larger domain comprises 52 colored kernel models and 8,265 decorated modules; it certifies 44,665,716 target obligations. We also classify the proper tight blocks at deficiency three, solve the resulting infinite corridor family, and reduce the unresolved three-missing-neighbor problem to at most one simple atomic 3-connected core along any chosen decomposition. The three-missing-neighbor theorem and general graphic subsequence-interchangeable base orderability remain open.