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
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)=2Θ(th−2). We also show that fH(t)=2Ω(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)=2Θ(t) if H is a clique, and fH(t)=2Θ(th−2) otherwise.Comment: 24 pages, 3 figure