Does reducing a graph (removing or replacing vertices or edges) without changing its chromatic number has a specific name?
Take this cactus graph as an example (although my question is about an arbitrary graph):
The edges with vertex of degree 1 could be removed without affecting the vertex chromatic number. I think something similar should be possible with cycles e.g. removing the two vertices in the bottom of the cactus should not affect its chromatic number.
Are there polynomial algorithms that removes edges to the extent that no more edge can be removed without affecting an arbitrary graph chromatic number? I would prefer not to reinvent the wheel.
If the answer is no, are there algorithms that remove an arbitrary number of edges without affecting an arbitrary graph chromatic number?
My goal is to simplify graphs before feeding them into other algorithms.
I will also appreciate references to relevant literature. Thank you!