How to choose good diagonals when partitioning an orthogonal polygon into rectangles?
Starting with 5,7,$f$ and $g$ is correct. In order to complete the maximum independent you can proceed as follows. First, delete the vertices that are reachable from an unmatched vertex via an alternating path (in your example 5,6,7,$e$,$f$,$g$). This leaves a balanced bipartite graph (that is, it has the same number of vertices on each side), and you can simply add one of the two sides to your independent set. In your example, you get the maximum independent set $\{1,2,3,4,5,7,f,g\}$ or $\{a,b,c,d,5,7,f,g\}$.