Lower Bound for Independence Covering in C4-free Graphs

An independent set in a graph G is a set S of pairwise non-adjacent vertices in G. A family 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 is an element of F such that I subset of S . Lokshtanov et al. (ACM Transactions on Algorithms, 2020) showed that graphs of degeneracy d admit k-independence covering families of size ((k(d+1))(k)) . 2(o(kd)) . 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. (ACM Transactions on Algorithms, 2020) 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,K- d+1 with d+1 vertices on both sides as a subgraph, called K-d+1,K-d+1-free graphs, are a frequently considered generalization of d-degenerate graphs. This motivates the question of whether K-d,K- 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,K-2 -free graphs (or equivalently C-4 -free graphs) do not admit k-independence covering families of size f(k)n(k/4-epsilon).