Complexity of cycle transverse matching problems

Authors:
Ross Churchley;Jing Huang;Xuding Zhu
Affiliations:
Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada;Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada;Department of mathematics, Zhejiang Normal University, Jinhua, Zhejiang, People's Republic of China
Venue:
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
Year:
2011

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Abstract

The stable transversal problem for a fixed graph H asks whether a graph contains a stable set that meets every induced copy of H in the graph. Stable transversal problems generalize several vertex partition problems and have been studied for various classes of graphs. Following a result of Farrugia, the stable transversal problem for each Cℓ with ℓ≥3 is NP-complete. In this paper, we study an ‘edge version' of these problems. Specifically, we investigate the problem of determining whether a graph contains a matching that meets every copy of H. We show that the problem for C3 is polynomial and for each Cℓ with ℓ≥4 is NP-complete. Our results imply that the stable transversal problem for each Cℓ with ℓ≥4 remains NP-complete when it is restricted to line graphs. We show by contrast that the stable transversal problem for C3, when restricted to line graphs, is polynomial.