5 research outputs found
Hitting forbidden induced subgraphs on bounded treewidth graphs
For a fixed graph , the -IS-Deletion problem asks, given a graph ,
for the minimum size of a set such that does
not contain 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 , the smallest function
such that -IS-Deletion can be solved in time assuming the Exponential Time Hypothesis (ETH), where and
denote the treewidth and the number of vertices of the input graph,
respectively.
We show that for every graph on
vertices, and that if 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
deviates slightly from a clique, the function suffers a sharp
jump: if is obtained from a clique of size by removing one edge, then
. We also show that
when , and this reduction answers an open question of Mi. Pilipczuk
[MFCS 2011] about the function for the subgraph version.
Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant
of the problem, where each vertex of is colored with some color from
and we require to hit only induced copies of with matching colors. In this
case, we determine, under the ETH, the function for every connected
graph on vertices: if the problem can be solved in polynomial
time; if , if is a clique, and otherwise.Comment: 24 pages, 3 figure