| Summary: | The distinguishing number of a graph $G$ is a symmetry related graph
invariant whose study started two decades ago. The distinguishing number $D(G)$
is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A
distinguishing $d$-coloring is a coloring $c:V(G)\rightarrow\{1,...,d\}$
invariant only under the trivial automorphism. In this paper, we introduce a
game variant of the distinguishing number. The distinguishing game is a game
with two players, the Gentle and the Rascal, with antagonist goals. This game
is played on a graph $G$ with a set of $d\in\mathbb N^*$ colors. Alternately,
the two players choose a vertex of $G$ and color it with one of the $d$ colors.
The game ends when all the vertices have been colored. Then the Gentle wins if
the coloring is distinguishing and the Rascal wins otherwise. This game leads
to define two new invariants for a graph $G$, which are the minimum numbers of
colors needed to ensure that the Gentle has a winning strategy, depending on
who starts. These invariants could be infinite, thus we start by giving
sufficient conditions to have infinite game distinguishing numbers. We also
show that for graphs with cyclic automorphisms group of prime odd order, both
game invariants are finite. After that, we define a class of graphs, the
involutive graphs, for which the game distinguishing number can be
quadratically bounded above by the classical distinguishing number. The
definition of this class is closely related to imprimitive actions whose blocks
have size $2$. Then, we apply results on involutive graphs to compute the exact
value of these invariants for hypercubes and even cycles. Finally, we study odd
cycles, for which we are able to compute the exact value when their order is
not prime. In the prime order case, we give an upper bound of $3$.
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