Given : G is undirected simple directed graph & some of the vertices are odd. and add 1 vertex V in the graph and connect that vertex to all the odd degree vertices , now new Graph we have to verify :
please Don't Get confused with Connected or Disconnected, because we have a theorem that says :
for any disconnected or connected graph if their exist exactly 2 odd number of vertices then their must be an edge between them . so by that we can say the number of odd degree vertices will always be even.
now we are adding vertex V and connecting it to all the odd degree vertices that will make their degree as even.Moreover, the degree of new vertex V will be even a/c to theorem (the number of odd degree vertices will always be even. )
so it will be Euler graph.