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Some results on total domination in direct products of graphs

Paul Dorbec;  Sylvain Gravier;  Sandi Klavžar;  Simon Spacapan — 2006

Discussiones Mathematicae Graph Theory

Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The domination number of direct products of graphs is also bounded from below.

How Long Can One Bluff in the Domination Game?

Boštan Brešar;  Paul Dorbec;  Sandi Klavžar;  Gašpar Košmrlj — 2017

Discussiones Mathematicae Graph Theory

The domination game is played on an arbitrary graph G by two players, Dominator and Staller. The game is called Game 1 when Dominator starts it, and Game 2 otherwise. In this paper bluff graphs are introduced as the graphs in which every vertex is an optimal start vertex in Game 1 as well as in Game 2. It is proved that every minus graph (a graph in which Game 2 finishes faster than Game 1) is a bluff graph. A non-trivial infinite family of minus (and hence bluff) graphs is established. minus graphs...

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