In the page 82 B(G) is defined like independent sets like you say but in the page 187 it other kind of B and it is define like Betti number and it is defined below
B(G) is the Betti number of the graph G of order n and size m having k components and it is defined as B(G)=m-n+k.
You can see on the page 187 of "Graph and Digraphs" 4 edition of G Chartrand and L. Lesniak.
Please can you explain what does B(G) mean and draw a graph.
Vertex Chromatic Numbers and Betti Numbers are investigated.