An independent set in a graph $G$ is a set $S$ of pairwise non-adjacent vertices in $G$. A family $\mathcal{F}$ of independent sets in $G$ is called a $k$-independence covering family if for every independent set $I$ in $G$ of size at most $k$, there exists an $S \in \mathcal{F}$ such that $I \subseteq S$. Lokshtanov et al. [ACM Transactions on Algorithms, 2018] showed that graphs of degeneracy $d$ admit $k$-independence covering families of size $\binom{k(d+1)}{k} \cdot 2^{o(kd)} \cdot \log n$, and used this result to design efficient parameterized algorithms for a number of problems, including STABLE ODD CYCLE TRANSVERSAL and STABLE MULTICUT. In light of the results of Lokshtanov et al. it is quite natural to ask whether even more general families of graphs admit $k$-independence covering families of size $f(k)n^{O(1)}$. Graphs that exclude a complete bipartite graph $K_{d+1,d+1}$ with $d+1$ vertices on both sides as a subgraph, called $K_{d+1,d+1}$-free graphs, are a frequently considered generalization of $d$-degenerate graphs. This motivates the question whether $K_{d,d}$-free graphs admit $k$-independence covering families of size $f(k,d)n^{O(1)}$. Our main result is a resounding "no" to this question -- specifically we prove that even $K_{2,2}$-free graphs (or equivalently $C_4$-free graphs) do not admit $k$-independence covering families of size $f(k)n^{\frac{k}{4}-\epsilon}$.

翻译：图 $G$ 中的独立集是指由 $G$ 中两两不相邻的顶点构成的集合 $S$。图 $G$ 中独立集构成的族 $\mathcal{F}$ 称为 $k$-独立覆盖族，若对于 $G$ 中任意大小不超过 $k$ 的独立集 $I$，存在 $S \in \mathcal{F}$ 使得 $I \subseteq S$。Lokshtanov 等人 [ACM Transactions on Algorithms, 2018] 证明，退化度为 $d$ 的图存在大小为 $\binom{k(d+1)}{k} \cdot 2^{o(kd)} \cdot \log n$ 的 $k$-独立覆盖族，并利用该结果设计了若干问题的高效参数化算法，包括稳定奇环横贯和稳定多割问题。鉴于 Lokshtanov 等人的成果，自然提出一个疑问：是否存在更一般的图族，其 $k$-独立覆盖族大小可表示为 $f(k)n^{O(1)}$？将完全二分图 $K_{d+1,d+1}$（两侧各含 $d+1$ 个顶点）作为子图排除的图，即 $K_{d+1,d+1}$-自由图，是 $d$-退化图的一种常见推广形式。这引出一个问题：$K_{d,d}$-自由图是否存在大小为 $f(k,d)n^{O(1)}$ 的 $k$-独立覆盖族？我们的主要结果是对此问题的明确否定——具体而言，我们证明即使 $K_{2,2}$-自由图（等价于 $C_4$-自由图）也不存在大小为 $f(k)n^{\frac{k}{4}-\epsilon}$ 的 $k$-独立覆盖族。