Check whether given degrees of vertices represent a Graph or Tree
Given the number of vertices and the degree of each vertex where vertex numbers are 1, 2, 3,…n. The task is to identify whether it is a graph or a tree. It may be assumed that the graph is connected.
Input : 5 2 3 1 1 1 Output : Tree Explanation : The input array indicates that vertex one has degree 2, vertex two has degree 3, vertices 3, 4 and 5 have degree 1. 1 / \ 2 3 / \ 4 5 Input : 3 2 2 2 Output : Graph 1 / \ 2 - 3
The degree of a vertex is given by the number of edges incident or leaving from it.
This can simply be done using the properties of trees like –
- Tree is connected and has no cycles while graphs can have cycles.
- Tree has exactly n-1 edges while there is no such constraint for graph.
- It is given that the input graph is connected. We need at least n-1 edges to connect n nodes.
If we take the sum of all the degrees, each edge will be counted twice. Hence, for a tree with n vertices and n – 1 edges, sum of all degrees should be 2 * (n – 1). Please refer Handshaking Lemma for details.
So basically we need to check if sum of all degrees is 2*(n-1) ore not.
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