Discussiones Mathematicae Graph Theory 26(3) (2006) 457-474
doi: 10.7151/dmgt.1338

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SOME RECENT RESULTS ON DOMINATION IN GRAPHS

Michael D. Plummer

Department of Mathematics
Vanderbilt University
Nashville, Tennessee 37240, USA
e-mail: michael.d.plummer@vanderbilt.edu

Abstract

In this paper, we survey some new results in four areas of domination in graphs, namely:

(1) the toughness and matching structure of graphs having domination number 3 and which are "critical" in the sense that if one adds any missing edge, the domination number falls to 2;

(2) the matching structure of graphs having domination number 3 and which are "critical" in the sense that if one deletes any vertex, the domination number falls to 2;

(3) upper bounds on the domination number of cubic graphs; and

(4) upper bounds on the domination number of graphs embedded in surfaces.

Keywords: domination, matching, toughness, cubic graph, triangulation, genus.

2000 Mathematics Subject Classification: 05C10, 05C69, 05C70.

References

[1]N. Ananchuen and M.D. Plummer, Some results related to the toughness of 3-domination-critical graphs, Discrete Math. 272 (2003) 5-15, doi: 10.1016/S0012-365X(03)00179-1.
[2]N. Ananchuen and M.D. Plummer, Some results related to the toughness of 3-domination critical graphs, II, Utilitas Math. (2006), (to appear).
[3]N. Ananchuen and M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1-13, doi: 10.1016/S0012-365X(03)00243-7.
[4]N. Ananchuen and M.D. Plummer, 3-factor-criticality in domination critical graphs, (2005), (submitted).
[5]N. Ananchuen and M.D. Plummer, Matchings in 3-vertex-critical graphs: the even case, Networks 45 (2005) 210-213, doi: 10.1002/net.20065.
[6]N. Ananchuen and M.D. Plummer, Matchings in 3-vertex-critical graphs: the odd case, 2005, (submitted).
[7]N. Ananchuen and M.D. Plummer, On the connectivity and matchings in 3-vertex-critical claw-free graphs, Utilitas Math. (2006), (to appear).
[8]R.C. Brigham, P.Z. Chinn and R.D. Dutton, A study of vertex domination critical graphs (Dept. of Math. Tech. Report M-2, Univ. of Central Florida, 1984).
[9]R.C. Brigham, P.Z. Chinn and R.D. Dutton, Vertex domination-critical graphs, Networks 18 (1988) 173-179, doi: 10.1002/net.3230180304.
[10]Y. Chen, F. Tian and B. Wei, The 3-domination-critical graphs with toughness one, Utilitas Math. 61 (2002) 239-253.
[11]M. Cropper, D. Greenwell, A.J.W. Hilton and A. Kostochka, The domination number of cubic Hamiltonian graphs, (2005), (submitted).
[12]M.H. de Carvalho, C.L. Lucchesi and U.S.R. Murty, On a conjecture of Lovász concerning bricks. I. The characteristic of a matching covered graph, J. Combin. Theory (B) 85 (2002) 94-136, doi: 10.1006/jctb.2001.2091.
[13]M.H. de Carvalho, C.L. Lucchesi and U.S.R. Murty, On a conjecture of Lovász concerning bricks. II. Bricks of finite characteristic, J. Combin. Theory (B) 85 (2002) 137-180, doi: 10.1006/jctb.2001.2092.
[14]J. Edmonds, L. Lovász and W.R. Pulleyblank, Brick decompositions and the matching rank of graphs, Combinatorica 2 (1982) 247-274, doi: 10.1007/BF02579233.
[15]O. Favaron, On k-factor-critical graphs, Discuss. Math. Graph Theory 16 (1996) 41-51, doi: 10.7151/dmgt.1022.
[16]O. Favaron, E. Flandrin and Z. Ryjácek, Factor-criticality and matching extension in DCT-graphs, Discuss. Math. Graph Theory 17 (1997) 271-278, doi: 10.7151/dmgt.1054.
[17]J. Fiedler, J. Huneke, R. Richter and N. Robertson, Computing the orientable genus of projective graphs, J. Graph Theory 20 (1995) 297-308, doi: 10.1002/jgt.3190200305.
[18]E. Flandrin, F. Tian, B. Wei and L. Zhang, Some properties of 3-domination-critical graphs, Discrete Math. 205 (1999) 65-76, doi: 10.1016/S0012-365X(99)00038-2.
[19]J. Fulman, Domination in vertex and edge critical graphs (Manuscript, Harvard Univ., 1992).
[20]J. Fulman, D. Hanson and G. MacGillivray, Vertex domination-critical graphs, Networks 25 (1995) 41-43, doi: 10.1002/net.3230250203.
[21]M. Garey and D. Johnson, Computers and Intractability, A Guide to the Theory of NP-Completeness, W.H. Freeman and Co. (San Francisco, 1979) 190.
[22]K. Kawarabayashi, A. Saito and M.D. Plummer, Domination in a graph with a 2-factor, J. Graph Theory (2006), (to appear), (Early view online at www.interscience.wiley.com, DOI 10.10021 jgt. 20142).
[23]A.V. Kostochka and B.Y. Stodolsky, On domination in connected cubic graphs, Communication: Discrete Math. 304 (2005) 45-50, doi: 10.1016/j.disc.2005.07.005.
[24]A.V. Kostochka and B.Y. Stodolsky, personal communication, November, 2005.
[25]A.V. Kostochka and B.Y. Stodolsky, An upper bound on the domination number of n-vertex connected cubic graphs, December, 2005, (submitted).
[26]M. Las Vergnas, A note on matchings in graphs, Colloque sur la Théorie des Graphes (Paris, 1974) Cahiers Centre Études Rech. Opér. 17 (1975) 257-260.
[27]G. Liu and Q. Yu, On n-edge-deletable and n-critical graphs, Bull. Inst. Combin. Appl. 24 (1998) 65-72.
[28]L. Lovász, Matching structure and the matching lattice, J. Combin. Theory (B) 43 (1987) 187-222, doi: 10.1016/0095-8956(87)90021-9.
[29]L. Lovász and M.D. Plummer, Matching Theory, Ann. Discrete Math. 29 (North-Holland, Amsterdam, 1986).
[30]L. Matheson and R. Tarjan, Dominating sets in planar graphs, European J. Combin. 17 (1996) 565-568, doi: 10.1006/eujc.1996.0048.
[31]S. Norine and R. Thomas, Generating bricks, preprint, 2005.
[32]S. Norine and R. Thomas, Minimal bricks, J. Combin. Theory (B) (2005), (to appear).
[33]M.D. Plummer and X. Zha, On certain spanning subgraphs of embeddings with applications to domination, 2005, (submitted).
[34]B. Reed, Paths, stars and the number three, Combin. Probab. Comput. 5 (1996) 277-295, doi: 10.1017/S0963548300002042.
[35]D.P. Sumner, 1-factors and anti-factor sets, J. London Math. Soc. 13 (1976) 351-359, doi: 10.1112/jlms/s2-13.2.351.
[36]D.P. Sumner and P. Blitch, Domination critical graphs, J. Combin. Theory (B) 34 (1983) 65-76, doi: 10.1016/0095-8956(83)90007-2.

Received 23 November 2005