Upper bounds for domination related parameters in graphs on surfaces

Abstract In this paper we give tight upper bounds on the total domination number, the weakly connected domination number and the connected domination number of a graph in terms of order and Euler characteristic. We also present upper bounds for the restrained bondage number, the total restrained bondage number and the restricted edge connectivity of graphs in terms of the orientable/nonorientable genus and maximum degree.

[1]  Lutz Volkmann,et al.  Total restrained bondage in graphs , 2013 .

[2]  Upper bounds for the domination subdivision and bondage numbers of graphs on topological surfaces , 2013 .

[3]  S. Louis Hakimi,et al.  On Computing a Conditional Edge-Connectivity of a Graph , 1988, Inf. Process. Lett..

[5]  Jaroslav Ivanĉo,et al.  The Weight of a Graph , 1992 .

[6]  Laura A. Sanchis.  On the number of edges in graphs with a given connected domination number , 2000, Discret. Math..

[7]  Abdol-Hossein Esfahanian,et al.  Generalized Measures of Fault Tolerance with Application to N-Cube Networks , 1989, IEEE Trans. Computers.

[8]  Note on the bondage number of graphs on topological surfaces , 2013, Australas. J Comb..

[9]  Stanislav Jendroľ,et al.  A KOTZIG TYPE THEOREM FOR NON-ORIENTABLE SURFACES , 2006 .

[10]  Johannes H. Hattingh,et al.  Restrained bondage in graphs , 2008, Discret. Math..

[11]  Laura A. Sanchis.  On the number of edges in graphs with a given weakly connected domination number , 2002, Discret. Math..

[12]  Alexandr V. Kostochka,et al.  Describing 3-paths in normal plane maps , 2013, Discret. Math..

[13]  Jun-Ming Xu,et al.  On Bondage Numbers of Graphs: A Survey with Some Comments , 2012, 1204.4010.

[14]  M. Jungerman,et al.  A CHARACTERIZATION OF UPPER-EMBEDDABLE GRAPHS , 1978 .

[15]  Tomaž Pisanski,et al.  Orientably simple graphs , 1987 .

[16]  Saul Stahl,et al.  Generalized Embedding Schemes , 1978, J. Graph Theory.

[17]  Lutz Volkmann,et al.  On the Roman Bondage Number of Planar Graphs , 2011, Graphs Comb..

[18]  Douglas F. Rall,et al.  A bound on the size of a graph with given order and bondage number , 1999, Discret. Math..

[19]  Laura A. Sanchis.  Relating the size of a connected graph to its total and restricted domination numbers , 2004, Discret. Math..

[20]  Saieed Akbari,et al.  A Note on the Roman Bondage Number of Planar Graphs , 2013, Graphs Comb..

[21]  Johannes H. Hattingh,et al.  On weakly connected domination in graphs , 1997, Discret. Math..

[22]  G. Ringel.  Map Color Theorem , 1974 .