Total domination in categorical products of graphs

• Autorzy: Douglas F. Rall
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

categorical product; open packing; total domination; submultiplicative; supermultiplicative

Kategorie tematyczne

• 05C05: Trees
• 05C70: Factorization, matching, partitioning, covering and packing
• 05C69: Dominating sets, independent sets, cliques

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2005
• Tom: 25
• Numer: 1-2
• Strony: 35-44

Daty

wydano

2005

otrzymano

2003-10-24

poprawiono

2004-04-19

Twórcy

autor

Douglas F. Rall

• Department of Mathematics, Furman University, Greenville, South Carolina 29613, USA

Bibliografia

• [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.

Identyfikatory

DOI

10.7151/dmgt.1257