1,250 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
Locating-total dominating sets in twin-free graphs: a conjecture
A total dominating set of a graph is a set of vertices of such
that every vertex of has a neighbor in . A locating-total dominating set
of is a total dominating set of with the additional property that
every two distinct vertices outside have distinct neighbors in ; that
is, for distinct vertices and outside , where denotes the open neighborhood of . A graph is twin-free if
every two distinct vertices have distinct open and closed neighborhoods. The
location-total domination number of , denoted , is the minimum
cardinality of a locating-total dominating set in . It is well-known that
every connected graph of order has a total dominating set of size at
most . We conjecture that if is a twin-free graph of order
with no isolated vertex, then . We prove the
conjecture for graphs without -cycles as a subgraph. We also prove that if
is a twin-free graph of order , then .Comment: 18 pages, 1 figur
Parameterized and approximation complexity of the detection pair problem in graphs
We study the complexity of the problem DETECTION PAIR. A detection pair of a
graph is a pair of sets of detectors with , the
watchers, and , the listeners, such that for every pair
of vertices that are not dominated by a watcher of , there is a listener of
whose distances to and to are different. The goal is to minimize
. This problem generalizes the two classic problems DOMINATING SET and
METRIC DIMENSION, that correspond to the restrictions and
, respectively. DETECTION PAIR was recently introduced by Finbow,
Hartnell and Young [A. S. Finbow, B. L. Hartnell and J. R. Young. The
complexity of monitoring a network with both watchers and listeners.
Manuscript, 2015], who proved it to be NP-complete on trees, a surprising
result given that both DOMINATING SET and METRIC DIMENSION are known to be
linear-time solvable on trees. It follows from an existing reduction by Hartung
and Nichterlein for METRIC DIMENSION that even on bipartite subcubic graphs of
arbitrarily large girth, DETECTION PAIR is NP-hard to approximate within a
sub-logarithmic factor and W[2]-hard (when parameterized by solution size). We
show, using a reduction to SET COVER, that DETECTION PAIR is approximable
within a factor logarithmic in the number of vertices of the input graph. Our
two main results are a linear-time -approximation algorithm and an FPT
algorithm for DETECTION PAIR on trees.Comment: 13 page
Location-domination in line graphs
A set of vertices of a graph is locating if every two distinct
vertices outside have distinct neighbors in ; that is, for distinct
vertices and outside , , where
denotes the open neighborhood of . If is also a dominating set (total
dominating set), it is called a locating-dominating set (respectively,
locating-total dominating set) of . A graph is twin-free if every two
distinct vertices of have distinct open and closed neighborhoods. It is
conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the
metric dimension and the determining number of a graph. Applied Mathematics and
Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning.
Locating-total dominating sets in twin-free graphs: a conjecture. The
Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any
twin-free graph without isolated vertices has a locating-dominating set of
size at most one-half its order and a locating-total dominating set of size at
most two-thirds its order. In this paper, we prove these two conjectures for
the class of line graphs. Both bounds are tight for this class, in the sense
that there are infinitely many connected line graphs for which equality holds
in the bounds.Comment: 23 pages, 2 figure
Locating-dominating sets and identifying codes in graphs of girth at least 5
Locating-dominating sets and identifying codes are two closely related
notions in the area of separating systems. Roughly speaking, they consist in a
dominating set of a graph such that every vertex is uniquely identified by its
neighbourhood within the dominating set. In this paper, we study the size of a
smallest locating-dominating set or identifying code for graphs of girth at
least 5 and of given minimum degree. We use the technique of vertex-disjoint
paths to provide upper bounds on the minimum size of such sets, and construct
graphs who come close to meet these bounds.Comment: 20 pages, 9 figure
Random subgraphs make identification affordable
An identifying code of a graph is a dominating set which uniquely determines
all the vertices by their neighborhood within the code. Whereas graphs with
large minimum degree have small domination number, this is not the case for the
identifying code number (the size of a smallest identifying code), which indeed
is not even a monotone parameter with respect to graph inclusion.
We show that every graph with vertices, maximum degree
and minimum degree , for some
constant , contains a large spanning subgraph which admits an identifying
code with size . In particular, if
, then has a dense spanning subgraph with identifying
code , namely, of asymptotically optimal size. The
subgraph we build is created using a probabilistic approach, and we use an
interplay of various random methods to analyze it. Moreover we show that the
result is essentially best possible, both in terms of the number of deleted
edges and the size of the identifying code
Harmonic response of the organ of corti: results for wave dispersion
Inner ear is a remarkable multiphysical system and its modelling is a great challenge. The approach used in this paper aims to reproduce physic with a realistic description of the radial cross section of the cochlea. A 2D‐section of the organ of Corti is fully described. Wavenumbers and corresponding modes of propagation are calculated taking into account passive structural responses. The study is extended to six cross sections of the organ of Corti and a large frequency bandwidth from 100 Hz to 3 kHz. Dispersion curves reveal the influence of fluid structure interactions with a dispersive behavior at high frequencies. Longitudinal mechanical coupling provides new interacting modes of propagation
Centroidal bases in graphs
We introduce the notion of a centroidal locating set of a graph , that is,
a set of vertices such that all vertices in are uniquely determined by
their relative distances to the vertices of . A centroidal locating set of
of minimum size is called a centroidal basis, and its size is the
centroidal dimension . This notion, which is related to previous
concepts, gives a new way of identifying the vertices of a graph. The
centroidal dimension of a graph is lower- and upper-bounded by the metric
dimension and twice the location-domination number of , respectively. The
latter two parameters are standard and well-studied notions in the field of
graph identification.
We show that for any graph with vertices and maximum degree at
least~2, . We discuss the
tightness of these bounds and in particular, we characterize the set of graphs
reaching the upper bound. We then show that for graphs in which every pair of
vertices is connected via a bounded number of paths,
, the bound being tight for paths and
cycles. We finally investigate the computational complexity of determining
for an input graph , showing that the problem is hard and cannot
even be approximated efficiently up to a factor of . We also give an
-approximation algorithm
Breaking down the link between luminous and dark matter in massive galaxies
We present a study on the clustering of a stellar mass selected sample of
galaxies with stellar masses M*>10^10Msol at redshifts 0.4<z<2.0, taken from
the Palomar Observatory Wide-field Infrared Survey. We examine the clustering
properties of these stellar mass selected samples as a function of redshift and
stellar mass, and find that galaxies with high stellar masses have a
progressively higher clustering strength than galaxies with lower stellar
masses. We also find that galaxies within a fixed stellar mass range have a
higher clustering strength at higher redshifts. We further estimate the average
total masses of the dark matter haloes hosting these stellar-mass selected
galaxies. For all galaxies in our sample the stellar-mass-to-total-mass ratio
is always lower than the universal baryonic mass fraction and the
stellar-mass-to-total-mass ratio is strongly correlated with the halo masses
for central galaxies, such that more massive haloes contain a lower fraction of
their mass in the form of stars. The remaining baryonic mass is included
partially in stars within satellite galaxies in these haloes, and as diffuse
hot and warm gas. We also find that, at a fixed stellar mass, the
stellar-to-total-mass ratio increases at lower redshifts. This suggests that
galaxies at a fixed stellar mass form later in lower mass dark matter haloes,
and earlier in massive haloes. We interpret this as a `halo downsizing' effect.Comment: Proceedings of the IAU Symposium No. 277, 2010 "Tracing the Ancestry
of Galaxies on the Land of our Ancestors"; Eds. Carignan, Freeman and Combe
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