The induced odd cycle packing number $iocp(G)$ of a graph $G$ is the maximum integer $k$ such that $G$ contains an induced subgraph consisting of $k$ pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al.~\cite{indoc} proved that graphs of bounded induced odd cycle packing number, bounded VC dimension, and linear independence number admit a randomized EPTAS for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm 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 assuming bounded induced odd cycle packing number.
翻译：以G$为单位的诱发奇周期包装编号为 $iocp(G) $G$是最大整数 $k$,因此,$G$包含一个诱发子子图,由对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对。通过几何图的应用,Bonamy et al. ⁇ cite{indoc} 证明,捆绑的奇周期包装编号、捆绑的VC维度和线性独立号的图表中包含一个随机的EPTAS。我们显示,对受约束的VC维维维维维值的假设没有必要,而对于任何整值为$kge 0美元和$t\ge 1$t\ g$ 和任何一美元正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对正对方方方方方方方方方方方方方方方对方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方
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