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 degG u + degG v ≥ 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 degG u + degG v and the ∗Partially supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 19500017, 2009. 144 J. Fujisawa, A. Saito and I. Schiermeyer structure of the distant area for u and v. We prove that if the distant area containsKr, we can relax the lower bound of degG u+degG 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.