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How Long Can One Bluff in the Domination Game?

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Brešar B., Dorbec P., Klavžar S., Košmrlj G.
Discussiones Mathematicae Graph Theory
|
2017
|
tom 37
|
nr 2
337-352
EN
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 with game domination number equal to 3 are characterized. Double bluff graphs are also introduced and it is proved that Kneser graphs K(n, 2), n ≥ 6, are double bluff. The domination game is also studied on generalized Petersen graphs and on Hamming graphs. Several generalized Petersen graphs that are bluff graphs but not vertex-transitive are found. It is proved that Hamming graphs are not double bluff.
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Some results on total domination in direct products of graphs

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Dorbec P., Gravier S., Klavžar S., Spacapan S.
Discussiones Mathematicae Graph Theory
|
2006
|
tom 26
|
nr 1
103-112
EN
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.
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