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Title:
Spanning Trees with few Branch and End Vertices
Other title:
Կմախքային ծառեր քիչ ճյուղային և կախված գագաթներով; Каркасы с менышим числом висячих и Br-вершин
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Uncontrolled Keywords:
Spanning tree ; End vertex ; k-ended tree ; Branch vertex ; Degree sum of the branch vertices
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Abstract:
For a graph G, let °² be the minimum degree sum of two nonadjacent vertices in G. A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. We consider: (*) connected graphs on n vertices such that °² ≥ n-k+1 for some positive integer k. In 1976, it was proved (by the author) that every graph satisfying (¤), has a spanning tree with at most k end vertices. In this paper we first show that every graph satisfying (*), has a spanning tree with at most k+1 branch and end vertices altogether. The next result states that every graph satisfying (¤), has a spanning tree with at most 1 2 (k-1) branch vertices. The third result states that every graph satisfying (*), has a spanning tree with at most 3 2 (k-1) degree sum of branch vertices. All results are sharp.