Lower Bound for Independence Covering in C_4-Free Graphs
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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 ∈ℱ such that I ⊆ S. Lokshtanov et al. [ACM Transactions on Algorithms, 2018] 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. 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^k/4-ϵ.
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