# Maximum independent set in "subgraph-claw-free" graphs

A $$d$$-claw in a graph is a set of $$d+1$$ vertices, one of which (the "center") is connected to the other $$d$$, but the other $$d$$ are not connected to each other. A graph is called $$d$$-claw free if it has no $$d$$-claw, that is: if one vertex is connected to $$d$$ other vertices, then at least two of these vertices are connected. One advantage of $$d$$-claw-free graphs is that they allow efficient approximation algorithms for the Maximum Independent Set problem. For example, this algorithm by Meike Neuwohner provides an approximation of about $$d/2$$.

Let us call a graph $$G$$ subgraph $$d$$-claw free if in every subgraph of $$G$$, there is at least one vertex that is "$$d$$-claw-free", that is, not the center of a $$d$$-claw. This property is weaker than $$d$$-claw-free.

In a subgraph $$d$$-claw free graph, one can start with an empty set, add a $$d$$-claw-free vertex, remove this vertex and all its neighbors, and repeat. Every added vertex invalidates at most $$d-1$$ vertices from the maximum independent set, so this algorithm achieves a $$(d-1)$$-approximation.

Question: is there a better approximation for the maximum independent set in subgraph $$d$$-claw free graphs?