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Saturday, November 3, 2018

Theorem: A simple graph is connected if and only if it has a spanning tree.

November 03, 2018  BCA-3rd Sem, Question Answer  No comments

Proof: Suppose that a simple graph G has a spanning tree T. Tcontains every vertex of G and there is a path in T between any  two of its vertices. Because T is a subgraph of G, there is a path in G between any two of its vertices. Hence, G is connected.  Now suppose that G is connected. If G is not a tree, it contains a simple circuit. Remove an edge from one of the simple circuits. The resulting subgraph is still connected because any vertices  connected via a path containing the removed edge are still connected via a path with the remaining part of the simple circuit. Continue in this fashion until there are no more simple circuits. A tree is produced because the graph remains connected as edges are removed. The resulting tree is a spanning tree because it contains every vertex of G. 

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