The weighted Caro–Wei obstruction conjecture for odd-cycle blowups

Let $G$ be a graph, let $d(v)$ be the degree of $v$, let $\alpha(G)$ be its independence number, and let $f:V(G)\to\mathbb R$ satisfy

$$f(v)\leq \frac{1}{d(v)}$$

for every $v\in V(G)$. A blowup of an odd cycle is a graph obtained from a cycle of length $2k+1$ by replacing its vertices with nonempty cliques, with adjacency between cliques corresponding to adjacency on the cycle.

Weighted Caro–Wei obstruction conjecture. Unless $G$ contains either a clique $K\subseteq V(G)$ such that

$$\sum_{v\in K}f(v)>1,$$

or a subgraph $H$ isomorphic to a blowup of a cycle of length $2k+1$ such that

$$\sum_{v\in V(H)}f(v)>k,$$

one has

$$\alpha(G)\geq\sum_{v\in V(G)}f(v).$$

This conjecture proposes that clique obstructions and blowups of odd cycles are essentially the only obstructions to the weighted independence-number bound. It is motivated by the examples discussed immediately before the conjecture, including blowups of $C_5$ and $C_7$; no resolution is supplied in the source.