Abstract
Finding dense spanning trees (DST) in unweighted graphs is a variation of the well studied minimum spanning tree problem (MST). We utilize established mathematical properties of extremal structures with the minimum sum of distances between vertices to formulate some general conditions on the sum of vertex degrees. We analyze the performance of various combinations of these degree sum conditions in finding dense spanning subtrees and apply our approach to practical examples. After briefly describing our algorithm we also show how it can be used on variations of DST, motivated by variations of MST. Our work provide some insights on the role of various degree sums in forming dense spanning trees and hopefully lay the foundation for finding fast algorithms or heuristics for related problems.
Highlights
ObjectivesNote that our goal is to analyze the effectiveness of each degree sum condition and we make no attempt in optimizing the algorithm
We explore the known mathematical properties of dense trees that lead to useful methods for solving dense spanning trees (DST)
In order to further explore the potential of using degrees as a credential for measuring the denseness of a spanning tree, we explore a number of conditions on the sum of vertex degrees
Summary
Note that our goal is to analyze the effectiveness of each degree sum condition and we make no attempt in optimizing the algorithm
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