Connected domination game played on Cartesian products

The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G . If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G . Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P 3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with K k for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.

Abstract licence: CC BY

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