459 research outputs found
Vertex identifying codes for the n-dimensional lattice
An -identifying code on a graph is a set such that for
every vertex in , the intersection of the radius- closed neighborhood
with is nonempty and different. Here, we provide an overview on codes for
the -dimensional lattice, discussing the case of 1-identifying codes,
constructing a sparse code for the 4-dimensional lattice as well as showing
that for fixed , the minimum density of an -identifying code is
.Comment: 10p
Perfect domination in regular grid graphs
We show there is an uncountable number of parallel total perfect codes in the
integer lattice graph of . In contrast, there is just one
1-perfect code in and one total perfect code in
restricting to total perfect codes of rectangular grid graphs (yielding an
asymmetric, Penrose, tiling of the plane). We characterize all cycle products
with parallel total perfect codes, and the -perfect and
total perfect code partitions of and , the former
having as quotient graph the undirected Cayley graphs of with
generator set . For , generalization for 1-perfect codes is
provided in the integer lattice of and in the products of cycles,
with partition quotient graph taken as the undirected Cayley graph
of with generator set .Comment: 16 pages; 11 figures; accepted for publication in Austral. J. Combi
Dissections, Hom-complexes and the Cayley trick
We show that certain canonical realizations of the complexes Hom(G,H) and
Hom_+(G,H) of (partial) graph homomorphisms studied by Babson and Kozlov are in
fact instances of the polyhedral Cayley trick. For G a complete graph, we then
characterize when a canonical projection of these complexes is itself again a
complex, and exhibit several well-known objects that arise as cells or
subcomplexes of such projected Hom-complexes: the dissections of a convex
polygon into k-gons, Postnikov's generalized permutohedra, staircase
triangulations, the complex dual to the lower faces of a cyclic polytope, and
the graph of weak compositions of an integer into a fixed number of summands.Comment: 23 pages, 5 figures; improved exposition; accepted for publication in
JCT
On Vertex Identifying Codes For Infinite Lattices
PhD Thesis--A compilation of the papers: "Lower Bounds for Identifying Codes
in Some Infinite Grids", "Improved Bounds for r-identifying Codes of the Hex
Grid", and "Vertex Identifying Codes for the n-dimensional Lattics" along with
some other resultsComment: 91p
An improved lower bound for (1,<=2)-identifying codes in the king grid
We call a subset of vertices of a graph a -identifying
code if for all subsets of vertices with size at most , the sets
are distinct. The concept of
identifying codes was introduced in 1998 by Karpovsky, Chakrabarty and Levitin.
Identifying codes have been studied in various grids. In particular, it has
been shown that there exists a -identifying code in the king grid
with density 3/7 and that there are no such identifying codes with density
smaller than 5/12. Using a suitable frame and a discharging procedure, we
improve the lower bound by showing that any -identifying code of
the king grid has density at least 47/111
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