5,143 research outputs found
A New Game Invariant of Graphs: the Game Distinguishing Number
The distinguishing number of a graph is a symmetry related graph
invariant whose study started two decades ago. The distinguishing number
is the least integer such that has a -distinguishing coloring. A
distinguishing -coloring is a coloring
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 with a set of colors. Alternately,
the two players choose a vertex of and color it with one of the 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 , 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 . 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
Bounds on the Game Transversal Number in Hypergraphs
Let be a hypergraph with vertex set and edge set of order
\nH = |V| and size \mH = |E|. A transversal in is a subset of vertices
in that has a nonempty intersection with every edge of . A vertex hits
an edge if it belongs to that edge. The transversal game played on involves
of two players, \emph{Edge-hitter} and \emph{Staller}, who take turns choosing
a vertex from . Each vertex chosen must hit at least one edge not hit by the
vertices previously chosen. The game ends when the set of vertices chosen
becomes a transversal in . Edge-hitter wishes to minimize the number of
vertices chosen in the game, while Staller wishes to maximize it. The
\emph{game transversal number}, , of is the number of vertices
chosen when Edge-hitter starts the game and both players play optimally. We
compare the game transversal number of a hypergraph with its transversal
number, and also present an important fact concerning the monotonicity of
, that we call the Transversal Continuation Principle. It is known that
if is a hypergraph with all edges of size at least~, and is not a
-cycle, then \tau_g(H) \le \frac{4}{11}(\nH+\mH); and if is a
(loopless) graph, then \tau_g(H) \le \frac{1}{3}(\nH + \mH + 1). We prove
that if is a -uniform hypergraph, then \tau_g(H) \le \frac{5}{16}(\nH +
\mH), and if is -uniform, then \tau_g(H) \le \frac{71}{252}(\nH +
\mH).Comment: 23 pages
Absorption Time of the Moran Process
The Moran process models the spread of mutations in populations on graphs. We
investigate the absorption time of the process, which is the time taken for a
mutation introduced at a randomly chosen vertex to either spread to the whole
population, or to become extinct. It is known that the expected absorption time
for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which
allows the behaviour of the process on undirected graphs to be analysed using
the Markov chain Monte Carlo method. We show that this does not extend to
directed graphs by exhibiting an infinite family of directed graphs for which
the expected absorption time is exponential in the number of vertices. However,
for regular directed graphs, we show that the expected absorption time is
Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds
and give improved bounds for other families of graphs, based on isoperimetric
number. Our results are obtained via stochastic dominations which we
demonstrate by establishing a coupling in a related continuous-time model. The
coupling also implies several natural domination results regarding the fixation
probability of the original (discrete-time) process, resolving a conjecture of
Shakarian, Roos and Johnson.Comment: minor change
Evolutionary Games on Networks and Payoff Invariance Under Replicator Dynamics
The commonly used accumulated payoff scheme is not invariant with respect to
shifts of payoff values when applied locally in degree-inhomogeneous population
structures. We propose a suitably modified payoff scheme and we show both
formally and by numerical simulation, that it leaves the replicator dynamics
invariant with respect to affine transformations of the game payoff matrix. We
then show empirically that, using the modified payoff scheme, an interesting
amount of cooperation can be reached in three paradigmatic non-cooperative
two-person games in populations that are structured according to graphs that
have a marked degree inhomogeneity, similar to actual graphs found in society.
The three games are the Prisoner's Dilemma, the Hawks-Doves and the Stag-Hunt.
This confirms previous important observations that, under certain conditions,
cooperation may emerge in such network-structured populations, even though
standard replicator dynamics for mixing populations prescribes equilibria in
which cooperation is totally absent in the Prisoner's Dilemma, and it is less
widespread in the other two games.Comment: 20 pages, 8 figures; to appear on BioSystem
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