Closure for spanning trees and distant area

A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with ${\displaystyle de{g}_{G}u+de{g}_{G}v\ge n-1}$, G has a spanning k-ended tree if and only if G+uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on ${\displaystyle de{g}_{G}u+de{g}_{G}v}$ and the structure of the distant area for u and v. We prove that if the distant area contains ${\displaystyle K_r}$, we can relax the lower bound of ${\displaystyle de{g}_{G}u+de{g}_{G}v}$ from n-1 to n-r. And if the distant area itself is a complete graph and G is 2-connected, we can entirely remove the degree sum condition.