A degree-constrained minimum spanning tree is a minimum spanning tree (MST) where the maximum vertex degree is limited to a certain constant k. The problem is NP-hard. Adapt Prim's and Kruskal's MST algorithms, and implement them, to find degree-constrained MST. Compare the resulting MSTs obtained from these two approximation techniques. In particular, first test your implementations first with a degree constraint of 5 or higher before becoming ambitious with lower values.

A spanning tree is a subset of Graph G, which has all the vertices covered with minimum possible number of edges. Hence, a spanning tree does not have cycles and it cannot be disconnected..

By this definition, we can draw a conclusion that every connected and undirected Graph G has at least one spanning tree. A disconnected graph does not have any spanning tree, as it cannot be spanned to all its vertices.

Spanning Trees

We found three spanning trees off one complete graph. A complete undirected graph can have maximum nn-2 number of spanning trees, where n is the number of nodes. In the above addressed example, n is 3, hence 33−2 = 3 spanning trees are possible.

General Properties of Spanning Tree

We now understand that one graph can have more than one spanning tree. Following are a few properties of the spanning tree connected to graph G −

A connected graph G can have more than one spanning tree.

All possible spanning trees of graph G, have the same number of edges and vertices.

The spanning tree does not have any cycle (loops).

Removing one edge from the spanning tree will make the graph disconnected, i.e. the spanning tree is minimally connected.

Adding one edge to the spanning tree will create a circuit or loop, i.e. the spanning tree is maximally acyclic.

Explanation:

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