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The induced odd cycle packing number $iocp(G)$ of a graph $G$ is the maximuminteger $k$ such that $G$ contains an induced subgraph consisting of $k$pairwise vertex-disjoint odd cycles. Motivated by applications to geometricgraphs, Bonamy et al.~\cite{indoc} proved that graphs of bounded induced oddcycle packing number, bounded VC dimension, and linear independence numberadmit a randomized EPTAS for the independence number. We show that theassumption of bounded VC dimension is not necessary, exhibiting a randomizedalgorithm that for any integers $k\ge 0$ and $t\ge 1$ and any $n$-vertex graph$G$ of induced odd cycle packing number at most $k$ returns in time$O_{k,t}(n^{k+4})$ an independent set of $G$ whose size is at least$\alpha(G)-n/t$ with high probability. In addition, we present$\chi$-boundedness results for graphs with bounded odd cycle packing number,and use them to design a QPTAS for the independence number only assumingbounded induced odd cycle packing number.

Induced odd cycle packing number, independent sets, and chromatic number