6,009 research outputs found
Enumerating planar locally finite Cayley graphs
We characterize the set of planar locally finite Cayley graphs, and give a
finite representation of these graphs by a special kind of finite state
automata called labeling schemes. As a result, we are able to enumerate and
describe all planar locally finite Cayley graphs of a given degree. This
analysis allows us to solve the problem of decision of the locally finite
planarity for a word-problem-decidable presentation.
Keywords: vertex-transitive, Cayley graph, planar graph, tiling, labeling
schemeComment: 19 pages, 6 PostScript figures, 12 embedded PsTricks figures. An
additional file (~ 438ko.) containing the figures in appendix might be found
at http://www.labri.fr/Perso/~renault/research/pages.ps.g
Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems
A complete classification of the computational complexity of the fixed-point
existence problem for boolean dynamical systems, i.e., finite discrete
dynamical systems over the domain {0, 1}, is presented. For function classes F
and graph classes G, an (F, G)-system is a boolean dynamical system such that
all local transition functions lie in F and the underlying graph lies in G. Let
F be a class of boolean functions which is closed under composition and let G
be a class of graphs which is closed under taking minors. The following
dichotomy theorems are shown: (1) If F contains the self-dual functions and G
contains the planar graphs then the fixed-point existence problem for (F,
G)-systems with local transition function given by truth-tables is NP-complete;
otherwise, it is decidable in polynomial time. (2) If F contains the self-dual
functions and G contains the graphs having vertex covers of size one then the
fixed-point existence problem for (F, G)-systems with local transition function
given by formulas or circuits is NP-complete; otherwise, it is decidable in
polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24
versio
Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems
We present dichotomy theorems regarding the computational complexity of
counting fixed points in boolean (discrete) dynamical systems, i.e., finite
discrete dynamical systems over the domain {0,1}. For a class F of boolean
functions and a class G of graphs, an (F,G)-system is a boolean dynamical
system with local transitions functions lying in F and graphs in G. We show
that, if local transition functions are given by lookup tables, then the
following complexity classification holds: Let F be a class of boolean
functions closed under superposition and let G be a graph class closed under
taking minors. If F contains all min-functions, all max-functions, or all
self-dual and monotone functions, and G contains all planar graphs, then it is
#P-complete to compute the number of fixed points in an (F,G)-system; otherwise
it is computable in polynomial time. We also prove a dichotomy theorem for the
case that local transition functions are given by formulas (over logical
bases). This theorem has a significantly more complicated structure than the
theorem for lookup tables. A corresponding theorem for boolean circuits
coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on
Theoretical Computer Science (ICTCS'2007
Causal Dynamics of Discrete Surfaces
We formalize the intuitive idea of a labelled discrete surface which evolves
in time, subject to two natural constraints: the evolution does not propagate
information too fast; and it acts everywhere the same.Comment: In Proceedings DCM 2013, arXiv:1403.768
On the Recognition of Families of Graphs with Local Computations
AbstractThis paper is a contribution to understanding the power and the limitations of local computations in graphs. We use local computations to define a notion of graph recognition; our model allows a simulation of automata on words and on trees. We introduce the notion of k-covering to examine limitations of such systems. For example, we prove that the family of series-parallel graphs and the family of planar graphs cannot be recognized by means of local computations
k-Colorability is Graph Automaton Recognizable
Automata operating on general graphs have been introduced by virtue of
graphoids. In this paper we construct a graph automaton that recognizes
-colorable graphs
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