Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration

We continue research into a well-studied family of problems that ask whether the vertices of a graph can be partitioned into sets $A$ and~$B$, where $A$ is an independent set and $B$ induces a graph from some specified graph class ${\cal G}$. We let ${\cal G}$ be the class of $k$-degenerate graphs.