Rooted dense minor linkage ($2\delta (H)\ge |H|+4k-1$)

If a graph is $2k$-connected and has a graph minor $H$ satisfying $2\delta (H)\ge |H|+4k-1$, it is $k$-linked. Write $d=|H|-1-\delta (H)$, so $|H|\ge 2d+4k+1$. The rooted branch-set reduction replaces the minor model by $|H|-k$ disjoint connected sets, the first $2k$ containing any prescribed terminals separately, each missing at most $d$ of the remaining sets. Every prescribed pair then has at least $k+1$ common adjacent unused sets, allowing the $k$ connections to be selected successively. The reduction uses rooted separations of a graph, contraction of edges outside distinct branch sets, and a final Hall marriage theorem matching; the separation condition prevents a contraction from destroying access to the model.