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Total Domination in Categorical Products of Graphs

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.