Minimum degree spanning tree explained

In graph theory, a minimum degree spanning tree is a subset of the edges of a connected graph that connects all the vertices together, without any cycles, and its maximum degree of its vertices as small as possible. That is, it is a spanning tree whose maximum degree is minimal.

The decision problem is: Given a graph G and an integer k, does G have a spanning tree such that no vertex has degree greater than k? This is also known as the degree-constrained spanning tree problem.

Algorithms

Finding the minimum degree spanning tree of an undirected graph is NP-hard. This can be shown by reduction to the Hamiltonian path problem. For directed graphs, finding the minimum degree spanning tree is also NP-hard. ^[1]

R. Krishman and B. Raghavachari (2001) have a quasi-polynomial time approximation algorithm to solve the problem for directed graphs.

M. Haque, Md. R. Uddin, and Md. A. Kashem (2007) found a linear time algorithm that can find the minimum degree spanning tree of series-parallel graphs with small degrees.^[2]

G. Yao, D. Zhu, H. Li, and S. Ma (2008) found a polynomial time algorithm that can find the minimum degree spanning tree of directed acyclic graphs.^[3]

Notes and References

1. Book: Krishnan . Radha . Raghavachari . Balaji . FST TCS 2001: Foundations of Software Technology and Theoretical Computer Science . The Directed Minimum-Degree Spanning Tree Problem . Lecture Notes in Computer Science . 2001 . 2245 . 232–243 . 10.1007/3-540-45294-X_20 . 978-3-540-43002-5 . https://link.springer.com/chapter/10.1007/3-540-45294-X_20.
2. Book: Haque . Mohammed Atiqul . Uddin . Md. Reaz . Kashem . Md. Abul . 2007 International Conference on Information and Communication Technology . An Algorithm for Finding Minimum Degree Spanning Tree of Series-Parallel Graphs . 2007 . 27–31 . 10.1109/ICICT.2007.375336 . 978-984-32-3394-3 . 17947444 . https://ieeexplore.ieee.org/document/4261359.
3. Yao . Guohui . Zhu . Daming . Li . Hengwu . Ma . Shaohan . A polynomial algorithm to compute the minimum degree spanning trees of directed acyclic graphs with applications to the broadcast problem . Discrete Mathematics . 6 September 2008 . 308 . 17 . 3951–3959 . 10.1016/j.disc.2007.07.105.