Applications of the splittance of a graph/ Turning graphs into splitgraphs
There is nothing deeper to it than what you describe: if the graph is split or it has &quot;small&quot; distance to being split (e.g., you can make it split by removing a small number of edges/vertices), you might be able to devise a polynomial-time algorithm for some otherwise NP-hard problem.
However, it is quite rare such procedures are applied in practice. It can take too much time to decide if the input graph has the desired properties compared to just using a state-of-the-art solver (e.g., integer programming) for the problem you care about. In fact, it is possible these solvers implicitly detect such useful structure and there is no need for this &quot;additional&quot; theoretical analysis.
To summarize, algorithms for split graphs and the kind of situations you describe are mostly theoretical and rarely implemented or used in practice.