Discussiones Mathematicae Graph Theory 25(1-2) (2005) 35-44
doi: 10.7151/dmgt.1257

[BIBTex] [PDF] [PS]


Douglas F. Rall

Department of Mathematics
Furman University
Greenville, South Carolina 29613, USA
e-mail: drall@herky.furman.edu


Several of the best known problems and conjectures in graph theory arise in studying the behavior of a graphical invariant on a graph product. Examples of this are Vizing's conjecture, Hedetniemi's conjecture and the calculation of the Shannon capacity of graphs, where the invariants are the domination number, the chromatic number and the independence number on the Cartesian, categorical and strong product, respectively. In this paper we begin an investigation of the total domination number on the categorical product of graphs. In particular, we show that the total domination number of the categorical product of a nontrivial tree and any graph without isolated vertices is equal to the product of their total domination numbers. In the process we establish a packing and covering equality for trees analogous to the well-known result of Meir and Moon. Specifically, we prove equality between the total domination number and the open packing number of any tree of order at least two.

Keywords: categorical product, open packing, total domination, submultiplicative, supermultiplicative.

2000 Mathematics Subject Classification: 05C69, 05C70, 05C05.


[1] B.D. Acharya, Graphs whose r-neighbourhoods form conformal hypergraphs, Indian J. Pure Appl. Math. 16 (5) (1985) 461-464.
[2] B.L. Hartnell and D.F. Rall, Lower bounds for dominating Cartesian products, J. Combin. Math. Combin. Comput. 31 (1999) 219-226.
[3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998).
[4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, Inc. New York, 1998).
[5] M.A. Henning, Packing in trees, Discrete Math. 186 (1998) 145-155, doi: 10.1016/S0012-365X(97)00228-8.
[6] M.A. Henning and D.F. Rall, On the total domination number of Cartesian products of graphs, Graphs and Combinatorics, to appear.
[7] M.A. Henning and P.J. Slater, Open packing in graphs, J. Combin. Math. Combin. Comput. 29 (1999) 3-16.
[8] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition (John Wiley & Sons, Inc. New York, 2000).
[9] L. Lovász, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-267, doi: 10.1016/0012-365X(72)90006-4.
[10] A. Meir and J.W. Moon, Relations between packing and covering numbers of a tree, Pacific J. Math. 61 (1975) 225-233.
[11] R.J. Nowakowski and D.F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53-79, doi: 10.7151/dmgt.1023.

Received 24 October 2003
Revised 19 April 2004