This paper studies a 4-approximation algorithm for k-prize collecting Steiner tree problems. This problem generalizes both k-minimum spanning tree problems and prize collecting Steiner tree problems. Our proposed algorithm employs two 2-approximation algorithms for k-minimum spanning tree problems and prize collecting Steiner tree problems. Also our algorithm framework can be applied to a speci...

Given a set of n disjoint line segments in the plane, we show that it is always possible to form a spanning tree of the endpoints of the segments, such that each line segment is an edge of the tree and the tree has no crossing edges. We also show that the maximum degree of a node in the minimum weight spanning tree formed by these line segments is seven, and that there exists a set of line segm...

In this paper we consider a spanning tree problem of uncertain network, which is a natural uncertain variation of the deterministic minimum spanning tree problem. By means of uncertainty theory, we propose three types of minimum spanning tree models for uncertain network, i.e., expected minimum spanning tree, α-minimum spanning tree, and distribution minimum spanning tree. Generally, we can the...

A drawing of a given (abstract) tree that is a minimum spanning tree of the vertex set is considered aesthetically pleasing. However, such a drawing can only exist if the tree has maximum degree at most 6. What can be said for trees of higher degree? We approach this question by supposing that a partition or covering of the tree by subtrees of bounded degree is given. Then we show that if the p...