TY - JOUR

T1 - Parameterized complexity of induced graph matching on claw-free graphs

AU - Hermelin, Danny

AU - Mnich, Matthias

AU - Van Leeuwen, Erik Jan

N1 - Funding Information:
An extended abstract of the results in this paper have appeared in the Proceedings of 20th European Symposium on Algorithms (ESA 2012) []. Partially supported by ERC StG project PAAl No. 259515. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement No. 631163.11.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - The Induced Graph Matching problem asks to find k disjoint induced subgraphs isomorphic to a given graph H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in k on claw-free graphs when H is a fixed connected graph, and even admits a polynomial kernel when H is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove W[1] -hardness of Induced Graph Matching on graphs excluding K1,4 as an induced subgraph, for any fixed complete graph H. In particular, we show that Independent Set is W[1] -hard on K1,4-free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or NP-complete, depending on the connectivity of H and the structure of G.

AB - The Induced Graph Matching problem asks to find k disjoint induced subgraphs isomorphic to a given graph H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in k on claw-free graphs when H is a fixed connected graph, and even admits a polynomial kernel when H is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove W[1] -hardness of Induced Graph Matching on graphs excluding K1,4 as an induced subgraph, for any fixed complete graph H. In particular, we show that Independent Set is W[1] -hard on K1,4-free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or NP-complete, depending on the connectivity of H and the structure of G.

KW - Algorithms and data structures

KW - Claw-free graphs

KW - Fixed-parameter tractability

KW - Induced matchings

UR - http://www.scopus.com/inward/record.url?scp=84896691879&partnerID=8YFLogxK

U2 - 10.1007/s00453-014-9877-5

DO - 10.1007/s00453-014-9877-5

M3 - Article

AN - SCOPUS:84896691879

SN - 0178-4617

VL - 70

SP - 513

EP - 560

JO - Algorithmica

JF - Algorithmica

IS - 3

ER -