Hitting forbidden induced subgraphs on bounded treewidth graphs
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For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ⊆ V(G) such that G∖ S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) · n^O(1) assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively. We show that f_H(t) = 2^O(t^h-2) for every graph H on h ≥ 3 vertices, and that f_H(t) = 2^O(t) if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^Θ(t^h-2). We also show that f_H(t) = 2^Ω(t^h) when H=K_h,h, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_C_4(t) for the subgraph version. Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h≤ 2 the problem can be solved in polynomial time; if h≥ 3, f_H(t) = 2^Θ(t) if H is a clique, and f_H(t) = 2^Θ(t^h-2) otherwise.
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