GATE | Gate IT 2008 | Question 26

G is a simple undirected graph. Some vertices of G are of odd degree. Add a node v to G and make it adjacent to each odd degree vertex of G. The resultant graph is sure to be
(A) regular
(B) Complete
(C) Hamiltonian
(D) Euler

Answer: (D)

Explanation: For a graph to be Euler graph all the degrees must be Even for all nodes. In any graph all the Odd degree nodes are connected with a node.
And number of Odd degree vertices should be even.
So degree of this new node will be Even and as a new edge is formed between this new node and all other nodes of Odd degree hence here is not a single node exists with degree Odd

=> Euler Graph

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