The bondage number of graphs on topological surfaces and Teschner's conjecture

Abstract The bondage number of a graph is the smallest number of its edges whose removal results in a graph having a larger domination number. We provide constant upper bounds for the bondage number of graphs on topological surfaces, and improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable genera of graphs. Also, we present stronger upper bounds for graphs with no triangles and graphs with the number of vertices larger than a certain threshold in terms of graph genera. This settles Teschner’s Conjecture in affirmative for almost all graphs. As an auxiliary result, we show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus.

[1]  Yuanqiu Huang.  Maximum genus of a graph in terms of its embedding properties , 2003, Discret. Math..

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

[3]  Andrei V. Gagarin,et al.  Upper bounds for the bondage number of graphs on topological surfaces , 2013, Discret. Math..

[4]  Michael S. Jacobson,et al.  The bondage number of a graph , 1991, Discret. Math..

[5]  Mike Develin,et al.  On the bondage number of planar and directed graphs , 2006, Discret. Math..

[6]  Jinjiang Yuan,et al.  Bondage number of planar graphs , 2000, Discret. Math..

[7]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[8]  William L. Kocay,et al.  Embeddings of Small Graphs on the Torus , 2001 .

[9]  Douglas F. Rall,et al.  Bounds on the bondage number of a graph , 1994, Discret. Math..

[10]  Jun-Ming Xu,et al.  On the complexity of the bondage and reinforcement problems , 2012, J. Complex..

[11]  C. Thomassen.  The Jordan-Scho¨nflies theorem and the classification of surfaces , 1992 .

[12]  Frank Harary,et al.  Domination alteration sets in graphs , 1983, Discret. Math..

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

[14]  Jia Huang.  An improved upper bound for the bondage number of graphs on surfaces , 2012, Discret. Math..

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

[16]  Frank Harary,et al.  Graph Theory , 2016 .

[17]  Gerhard Ringel,et al.  The combinatorial map color theorem , 1977, J. Graph Theory.

[18]  H. Finck,et al.  Einführung in die Theorie der endlichen Graphen , 1971 .

[19]  Ulrich Teschner,et al.  A new upper bound for the bondage number of graphs with small domination number , 1995, Australas. J Comb..

[20]  Carsten Thomassen,et al.  The Graph Genus Problem is NP-Complete , 1989, J. Algorithms.

[21]  Donald L. Kreher,et al.  Graphs, algorithms and optimization , 2004 .

[22]  Dieter Rautenbach,et al.  Remarks on the bondage number of planar graphs , 2003, Discret. Math..

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