8,323 research outputs found
Bounds for identifying codes in terms of degree parameters
An identifying code is a subset of vertices of a graph such that each vertex
is uniquely determined by its neighbourhood within the identifying code. If
\M(G) denotes the minimum size of an identifying code of a graph , it was
conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there
exists a constant such that if a connected graph with vertices and
maximum degree admits an identifying code, then \M(G)\leq
n-\tfrac{n}{d}+c. We use probabilistic tools to show that for any ,
\M(G)\leq n-\tfrac{n}{\Theta(d)} holds for a large class of graphs
containing, among others, all regular graphs and all graphs of bounded clique
number. This settles the conjecture (up to constants) for these classes of
graphs. In the general case, we prove \M(G)\leq n-\tfrac{n}{\Theta(d^{3})}.
In a second part, we prove that in any graph of minimum degree and
girth at least 5, \M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n.
Using the former result, we give sharp estimates for the size of the minimum
identifying code of random -regular graphs, which is about
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
Identifying codes in vertex-transitive graphs and strongly regular graphs
We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs
Improved Bounds for -Identifying Codes of the Hex Grid
For any positive integer , 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 pairwise distinct. For
a finite graph, the density of a code is , which naturally extends
to a definition of density in certain infinite graphs which are locally finite.
We find a code of density less than , which is sparser than the prior
best construction which has density approximately .Comment: 12p
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