## Paired- and Induced Paired-domination in {E, net}-free Graphs

 Oliver Schaudt Institut für Informatik Universität zu Köln Weyertal 80, 50931 Cologne, Germany

## Abstract

A dominating set of a graph is a vertex subset that any vertex belongs to or is adjacent to. Among the many well-studied variants of domination are the so-called paired-dominating sets. A paired-dominating set is a dominating set whose induced subgraph has a perfect matching. In this paper, we continue their study.

We focus on graphs that do not contain the net-graph (obtained by attaching a pendant vertex to each vertex of the triangle) or the E-graph (obtained by attaching a pendant vertex to each vertex of the path on three vertices) as induced subgraphs. This graph class is a natural generalization of {claw, net}-free graphs, which are intensively studied with respect to their nice properties concerning domination and hamiltonicity. We show that any connected {E, net}-free graph has a paired-dominating set that, roughly, contains at most half of the vertices of the graph. This bound is a significant improvement to the known general bounds.

Further, we show that any {E, net, C5}-free graph has an induced paired-dominating set, that is a paired-dominating set that forms an induced matching, and that such set can be chosen to be a minimum paired-dominating set. We use these results to obtain a new characterization of {E, net, C5}-free graphs in terms of the hereditary existence of induced paired-dominating sets. Finally, we show that the induced matching formed by an induced paired-dominating set in a {E, net, C5}-free graph can be chosen to have at most two times the size of the smallest maximal induced matching possible.

Keywords: domination, paired-domination, induced paired-domination, induced matchings, { E, net}-free graphs

2010 Mathematics Subject Classification: 05C69.

## References

 [1] G. Bacsó, Complete description of forbidden subgraphs in the structural domination problem, Discrete Math. 309 (2009) 2466--2472, doi: 10.1016/j.disc.2008.05.053. [2] A. Brandstädt and F.F. Dragan, On linear and circular structure of (claw, net)-free graphs, Discrete Appl. Math. 129 (2003) 285--303, doi: 10.1016/S0166-218X(02)00571-1. [3] A. Brandstädt, F.F. Dragan and E. Köhler, Linear time algorithms for Hamiltonian problems on ( claw, net)-free graphs, SIAM J. Comput. 30 (2000) 1662--1677, doi: 10.1137/S0097539799357775. [4] K. Cameron, Induced matchings, Discrete Appl. Math. 24 (1989) 97--102, doi: 10.1016/0166-218X(92)90275-F. [5] P. Damaschke, Hamiltonian-hereditary graphs, manuscript (1990). [6] P. Dorbec and S. Gravier, Paired-domination in subdivided star-free graphs, Graphs Combin. 26 (2010) 43--49, doi: 10.1007/s00373-010-0893-1. [7] G. Finke, V. Gordon, Y.L. Orlovich and I.É. Zverovich, Approximability results for the maximum and minimum maximal induced matching problems, Discrete Optimization 5 (2008) 584--593, doi: 10.1016/j.disopt.2007.11.010. [8] D.L. Grinstead, P.J. Slater, N.A. Sherwani and N.D. Holmes, Efficient edge domination problems in graphs, Inform. Process. Lett. 48 (1993) 221--228, doi: 10.1016/0020-0190(93)90084-M. [9] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker New York, 1998). [10] T.W. Haynes, L.M. Lawson and D.S. Studer, Induced-paired domination in graphs, Ars Combin. 57 (2000) 111--128. [11] T.W. Haynes and P.J. Slater, Paired-domination in graphs, Networks 32 (1998) 199--206, doi: 10.1002/(SICI)1097-0037(199810)32:3<199::AID-NET4>3.0.CO;2-F. [12] A. Kelmans, On Hamiltonicity of {claw, net}-free graphs, Discrete Math. 306 (2006) 2755--2761, doi: 10.1016/j.disc.2006.04.022. [13] C.M. Mynhardt and M. Schurch, Paired domination in prisms of graphs, Discuss. Math. Graph Theory 31 (2011) 5--23, doi: 10.7151/dmgt.1526. [14] Y.L. Orlovich and I.É. Zverovich, Maximal induced matchings of minimum/maximum size, manuscript (2004). [15] O. Schaudt, Total domination versus paired-domination, Discuss. Math. Graph Theory 32 (2012) 435--447, doi: 10.7151/dmgt.1614. [16] O. Schaudt, On weighted efficient total domination, J. Discrete Algorithms 10 (2012) 61--69, doi: 10.1016/j.jda.2011.06.001. [17] J.A. Telle, Complexity of domination-type problems in graphs, Nordic J. Comput. 1 (1994) 157--171. [18] Z. Tuza, Hereditary domination in graphs: Characterization with forbidden induced subgraphs, SIAM J. Discrete Math. 22 (2008) 849--853, doi: 10.1137/070699482. [19] B. Zelinka, Induced-paired domatic numbers of graphs, Math. Bohem. 127 (2002) 591--596.