15,267 research outputs found
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
Maker-Breaker domination number
The Maker-Breaker domination game is played on a graph by Dominator and
Staller. The players alternatively select a vertex of that was not yet
chosen in the course of the game. Dominator wins if at some point the vertices
he has chosen form a dominating set. Staller wins if Dominator cannot form a
dominating set. In this paper we introduce the Maker-Breaker domination number
of as the minimum number of moves of Dominator to
win the game provided that he has a winning strategy and is the first to play.
If Staller plays first, then the corresponding invariant is denoted
. Comparing the two invariants it turns out that they
behave much differently than the related game domination numbers. The invariant
is also compared with the domination number. Using the
Erd\H{o}s-Selfridge Criterion a large class of graphs is found for which
holds. Residual graphs are introduced and
used to bound/determine and .
Using residual graphs, and are
determined for an arbitrary tree. The invariants are also obtained for cycles
and bounded for union of graphs. A list of open problems and directions for
further investigations is given.Comment: 20 pages, 5 figure
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