Upper bounds on the broadcast function using minimum dominating sets

Abstract We construct new upper bounds on the broadcast function B ( n ) , the number of edges in a minimum broadcast graph on n vertices, for a large class of integers n . The bounds are obtained by a construction that uses the minimum size of dominating sets for some Knodel graphs. We also determine the domatic number of these graphs.

[1]  André Raspaud,et al.  A survey on Knödel graphs , 2004, Discret. Appl. Math..

[2]  S. Hedetniemi,et al.  Domination in graphs : advanced topics , 1998 .

[3]  David Peleg,et al.  Tight Bounds on Minimum Broadcast Networks , 1991, SIAM J. Discret. Math..

[4]  Arthur L. Liestman,et al.  A survey of gossiping and broadcasting in communication networks , 1988, Networks.

[5]  Stephen T. Hedetniemi,et al.  Towards a theory of domination in graphs , 1977, Networks.

[6]  Hovhannes A. Harutyunyan,et al.  On the minimum path problem in Knödel graphs , 2007 .

[7]  Andrzej Pelc,et al.  Dissemination of Information in Communication Networks - Broadcasting, Gossiping, Leader Election, and Fault-Tolerance , 2005, Texts in Theoretical Computer Science. An EATCS Series.

[8]  Jose A. Ventura,et al.  Compound Constructions of Broadcast Networks , 1999, Discret. Appl. Math..

[9]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[10]  Claude Berge,et al.  The theory of graphs and its applications , 1962 .

[11]  Arthur L. Liestman,et al.  More Broadcast Graphs , 1999, Discret. Appl. Math..

[12]  Roger Labahn.  A minimum broadcast graph on 63 vertices , 1994, Discret. Appl. Math..

[13]  Pierre Fraigniaud,et al.  Minimum linear gossip graphs and maximal linear (Delta, k)-gossip graphs , 2001, Networks.

[14]  Hovhannes A. Harutyunyan.  Minimum multiple message broadcast graphs , 2006 .

[15]  Walter Knödel,et al.  New gossips and telephones , 1975, Discret. Math..

[16]  Stéphane Pérennes,et al.  A Note on the Dimensionality of Modified Knö;del Graphs , 1997, Int. J. Found. Comput. Sci..

[17]  Pierre Fraigniaud,et al.  Methods and problems of communication in usual networks , 1994, Discret. Appl. Math..

[18]  Arthur L. Liestman,et al.  On the monotonicity of the broadcast function , 2003, Discret. Math..

[19]  Hovhannes A. Harutyunyan.  Multiple message broadcasting in modified Knödel graph , 2000, SIROCCO.

[20]  Stéphane Pérennes,et al.  Complete Rotations in Cayley Graphs , 2001, Eur. J. Comb..

[21]  Arthur L. Liestman,et al.  Improved upper and lower bounds for k-broadcasting , 2001 .

[22]  Arthur L. Liestman,et al.  Sparse broadcast graphs , 1992, Discret. Appl. Math..

[23]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[24]  Hovhannes A. Harutyunyan,et al.  The spectra of Knödel graphs , 2006, Informatica.

[25]  Hovhannes A. Harutyunyan.  An Efficient Vertex Addition Method for Broadcast Networks , 2008, Internet Math..