Parameterized Complexity of Finding Subgraphs with Hereditary Properties

We consider the parameterized complexity of the following problem under the framework introduced by Downey and Fellows[4]: Given a graph G, an integer parameter k and a non-trivial hereditary property Π, are there k vertices of G that induce a subgraph with property Π? This problem has been proved NP-hard by Lewis and Yannakakis[9]. We show that if Π includes all independent sets but not all cliques or vice versa, then the problem is hard for the parameterized class W [1] and is fixed parameter tractable otherwise. In the former case, if the forbidden set of the property is finite, we show, in fact, that the problem is W [1]-complete (see [4] for definitions). Our proofs, both of the tractability as well as the hardness ones, involve clever use of Ramsey numbers.

• Type: Conference
• Year: 2000
• Where: COCOON
• Authors: Subhash Khot, Venkatesh Raman
• Keywords: COCOON 2000, Combinatorics, Non-trivial Hereditary Property, Parameterized Complexity, Property Π