22,163 research outputs found
The Complexity of Node Blocking for Dags
We consider the following modification of annihilation game called node
blocking. Given a directed graph, each vertex can be occupied by at most one
token. There are two types of tokens, each player can move his type of tokens.
The players alternate their moves and the current player selects one token
of type and moves the token along a directed edge to an unoccupied vertex.
If a player cannot make a move then he loses. We consider the problem of
determining the complexity of the game: given an arbitrary configuration of
tokens in a directed acyclic graph, does the current player has a winning
strategy? We prove that the problem is PSPACE-complete.Comment: 7 pages, 3 figure
Degree Sequence Index Strategy
We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by
which to bound graph invariants by certain indices in the ordered degree
sequence. As an illustration of the DSI strategy, we show how it can be used to
give new upper and lower bounds on the -independence and the -domination
numbers. These include, among other things, a double generalization of the
annihilation number, a recently introduced upper bound on the independence
number. Next, we use the DSI strategy in conjunction with planarity, to
generalize some results of Caro and Roddity about independence number in planar
graphs. Lastly, for claw-free and -free graphs, we use DSI to
generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
Static Pairwise Annihilation in Complex Networks
We study static annihilation on complex networks, in which pairs of connected
particles annihilate at a constant rate during time. Through a mean-field
formalism, we compute the temporal evolution of the distribution of surviving
sites with an arbitrary number of connections. This general formalism, which is
exact for disordered networks, is applied to Kronecker, Erd\"os-R\'enyi (i.e.
Poisson) and scale-free networks. We compare our theoretical results with
extensive numerical simulations obtaining excellent agreement. Although the
mean-field approach applies in an exact way neither to ordered lattices nor to
small-world networks, it qualitatively describes the annihilation dynamics in
such structures. Our results indicate that the higher the connectivity of a
given network element, the faster it annihilates. This fact has dramatic
consequences in scale-free networks, for which, once the ``hubs'' have been
annihilated, the network disintegrates and only isolated sites are left.Comment: 7 Figures, 10 page
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