## Total Domination Versus Paired Domination

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

## Abstract

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γt. The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Γt. A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by γp. The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Γp.

In this paper we prove several results on the ratio of these four parameters: For each r ≥ 2 we prove the sharp bound γp/ γt ≤ 2 − 2/r for K1,r-free graphs. As a consequence, we obtain the sharp bound γp/ γt ≤ 2 − 2/( Δ+1), where Δ is the maximum degree. We also show for each r ≥ 2 that {C5,Tr}-free graphs fulfill the sharp bound γp/ γt ≤ 2 − 2/r, where Tr is obtained from K1,r by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Γp / Γt. Further, we prove that a graph hereditarily has an induced paired dominating set if and only if γp ≤ Γt holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio γp / Γt taken over the induced subgraphs of a graph. As a consequence, we prove for each r ≥ 3 the sharp bound γp/ Γt ≤ 2 − 2/r for graphs that do not contain the corona of K1,r as subgraph. In particular, we obtain the sharp bound γp/ Γt ≤ 2 − 2/ Δ.

Keywords: total domination, upper total domination, paired domination, upper paired domination, generalized claw-free graphs

2010 Mathematics Subject Classification: 05C69.

## References

 [1] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, (Marcel Dekker, New York, 1998). [2] E.J. Cockayne, R.M. Dawes and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211--219, doi: 10.1002/net.3230100304. [3] M.A. Henning, A survey of selected recent results on total domination in graphs, Discrete Math. 309 (2009) 32--63, doi: 10.1016/j.disc.2007.12.044. [4] 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. [5] J.A. Telle, Complexity of domination-type problems in graphs, Nordic J. Comput. 1 (1994) 157--171. [6] T.W. Haynes, L.M. Lawson and D.S. Studer, Induced-paired domination in graphs, Ars Combin. 57 (2000) 111--128. [7] B. Zelinka, Induced-paired domatic numbers of graphs, Math. Bohem. 127 (2002) 591--596. [8] O. Favaron and M.A. Henning, Total domination in claw-free graphs with minimum degree 2, Discrete Math. 308 (2008) 3213--3219, doi: 10.1016/j.disc.2007.06.024. [9] O. Favaron and M.A. Henning, Bounds on total domination in claw-free cubic graphs, Discrete Math. 308 (2008) 3491--3507, doi: 10.1016/j.disc.2007.07.007. [10] O. Favaron and M.A. Henning, Upper total domination in claw-free graphs, J. Graph Theory 44 (2003) 148--158, doi: 10.1002/jgt.10134. [11] M.A. Henning and J. Southey, On a conjecture on total domination in claw-free cubic graphs, Discrete Math. 310 (2010) 2984--2999, doi: 10.1016/j.disc.2010.07.006. [12] P. Dorbec, S. Gravier and M.A. Henning, Paired-domination in generalized claw-free graphs, J. Comb. Optim. 14 (2007) 1--7, doi: 10.1007/s10878-006-9022-8. [13] 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. [14] M. Blidia, M. Chellali and O. Favaron, Ratios of Some Domination Parameters in Graphs and Claw-free Graphs, In: A. Bondy, J. Fonlupt, J.-L. Fouquet, J.-C. Fournier and J.L. Ramírez Alfonsín (Eds.), Trends in Mathematics: Graph Theory in Paris, Birkhäuser, Basel, (2007) 61?72, doi: 10.1007/978-3-7643-7400-6_6. [15] R.C. Brigham and R.D. Dutton, Domination in claw-free graphs, Congr. Numer. 132 (1998) 69--75. [16] P. Dorbec, M.A. Henning and J. McCoy, Upper total domination versus upper paired-domination, Quaest. Math. 30 (2007) 1--12, doi: 10.2989/160736007780205693. [17] O. Schaudt, On the existence of total dominating subgraphs with a prescribed additive hereditary property, Discrete Math. 311 (2011) 2095--2101, doi: 10.1016/j.disc.2011.05.036. [18] C. Berge, Two theorems in graph theory, Proceedings of the National Academy of Sciences of the United States of America 43 (1957) 842--844, doi: 10.1073/pnas.43.9.842. [19] M.D. Plummer and A. Saito, Forbidden subgraphs and bounds on the size of a maximum matching, J. Graph Theory 50 (2005) 1--12, doi: 10.1002/jgt.20087.