Hitting forbidden induced subgraphs on bounded treewidth graphs
Abstract
For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ⊂eq 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 fH(t) such that H-IS-Deletion can be solved in time fH(t) · nO(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 fH(t) = 2O(th-2) for every graph H on h ≥ 3 vertices, and that fH(t) = 2O(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 fH(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then fH(t) = 2(th-2). We also show that fH(t) = 2(th) when H=Kh,h, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function fC4(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 fH(t) for every connected graph H on h vertices: if h≤ 2 the problem can be solved in polynomial time; if h≥ 3, fH(t) = 2(t) if H is a clique, and fH(t) = 2(th-2) otherwise.