Q:

Genus of The Complete Bipartite Graph

Let $G=K_{3,7}$ be a complete bipartite graph with partitions $X=\{x_1,x_2,x_3\}$ and $Y=\{y_1,y_2,y_3,y_4,y_5,y_6,y_7\}$. We know that the genus of graph $G$ is $2$. If we delete three edges $\{x_1y_1\}$, $\{x_2y_2\}$ and $\{x_3y_3\}$ of graph $G$ and add three edges $\{x_1x_2\}$, $\{x_2x_3\}$ and $\{x_1x_3\}$ to graph $G$, then the genus of the resulting graph is again $2$?

A:

SageMath math says the genus is $1$.
sage: G = Graph([[0,1],[0,2],[1,2],[0,4],[0,5],[0,6],[0,7],[0,8],[0,9],[1,3],[1,5],[1,6],[1,7],[1,8],[1,9],[2,3],[2,4],[2,6],[2,7],[2,8],[2,9]])
sage: G.genus()
1
sage: G.get_embedding()

{0: [1, 2, 4, 5, 6, 7, 8, 9],
 1: [0, 9, 2, 3, 6, 5, 8, 7],
 2: [0, 7, 6, 3, 1, 9, 8, 4],
 3: [1, 2],
 4: [0, 2],
 5: [0, 1],
 6: [0, 1, 2],
 7: [0, 2, 1],
 8: [0, 1, 2],
 9: [0, 2, 1]} 

