445 research outputs found
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
Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity
We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) Open Locating-Dominating Set and Metric Dimension) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter 2 and permutation graphs of diameter 2. While Identifying Code and (Open) Locating-Dominating Set are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting Metric Dimension is W[2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable
On the Distance Identifying Set Meta-Problem and Applications to the Complexity of Identifying Problems on Graphs
Numerous problems consisting in identifying vertices in graphs using
distances are useful in domains such as network verification and graph
isomorphism. Unifying them into a meta-problem may be of main interest. We
introduce here a promising solution named Distance Identifying Set. The model
contains Identifying Code (IC), Locating Dominating Set (LD) and their
generalizations -IC and -LD where the closed neighborhood is considered
up to distance . It also contains Metric Dimension (MD) and its refinement
-MD in which the distance between two vertices is considered as infinite if
the real distance exceeds . Note that while IC = 1-IC and LD = 1-LD, we have
MD = -MD; we say that MD is not local
In this article, we prove computational lower bounds for several problems
included in Distance Identifying Set by providing generic reductions from
(Planar) Hitting Set to the meta-problem. We mainly focus on two families of
problem from the meta-problem: the first one, called bipartite gifted local,
contains -IC, -LD and -MD for each positive integer while the
second one, called 1-layered, contains LD, MD and -MD for each positive
integer . We have:
- the 1-layered problems are NP-hard even in bipartite apex graphs,
- the bipartite gifted local problems are NP-hard even in bipartite planar
graphs,
- assuming ETH, all these problems cannot be solved in when
restricted to bipartite planar or apex graph, respectively, and they cannot be
solved in on bipartite graphs,
- even restricted to bipartite graphs, they do not admit parameterized
algorithms in except if W[0] = W[2]. Here is the
solution size of a relevant identifying set.
In particular, Metric Dimension cannot be solved in under ETH,
answering a question of Hartung in 2013
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
Optimisation problems in wireless sensor networks : Local algorithms and local graphs
This thesis studies optimisation problems related to modern large-scale distributed systems, such as wireless sensor networks and wireless ad-hoc networks. The concrete tasks that we use as motivating examples are the following: (i) maximising the lifetime of a battery-powered wireless sensor network, (ii) maximising the capacity of a wireless communication network, and (iii) minimising the number of sensors in a surveillance application. A sensor node consumes energy both when it is transmitting or forwarding data, and when it is performing measurements. Hence task (i), lifetime maximisation, can be approached from two different perspectives. First, we can seek for optimal data flows that make the most out of the energy resources available in the network; such optimisation problems are examples of so-called max-min linear programs. Second, we can conserve energy by putting redundant sensors into sleep mode; we arrive at the sleep scheduling problem, in which the objective is to find an optimal schedule that determines when each sensor node is asleep and when it is awake. In a wireless network simultaneous radio transmissions may interfere with each other. Task (ii), capacity maximisation, therefore gives rise to another scheduling problem, the activity scheduling problem, in which the objective is to find a minimum-length conflict-free schedule that satisfies the data transmission requirements of all wireless communication links. Task (iii), minimising the number of sensors, is related to the classical graph problem of finding a minimum dominating set. However, if we are not only interested in detecting an intruder but also locating the intruder, it is not sufficient to solve the dominating set problem; formulations such as minimum-size identifying codes and locating–dominating codes are more appropriate. This thesis presents approximation algorithms for each of these optimisation problems, i.e., for max-min linear programs, sleep scheduling, activity scheduling, identifying codes, and locating–dominating codes. Two complementary approaches are taken. The main focus is on local algorithms, which are constant-time distributed algorithms. The contributions include local approximation algorithms for max-min linear programs, sleep scheduling, and activity scheduling. In the case of max-min linear programs, tight upper and lower bounds are proved for the best possible approximation ratio that can be achieved by any local algorithm. The second approach is the study of centralised polynomial-time algorithms in local graphs – these are geometric graphs whose structure exhibits spatial locality. Among other contributions, it is shown that while identifying codes and locating–dominating codes are hard to approximate in general graphs, they admit a polynomial-time approximation scheme in local graphs
Metric dimension for random graphs
The metric dimension of a graph is the minimum number of vertices in a
subset of the vertex set of such that all other vertices are uniquely
determined by their distances to the vertices in . In this paper we
investigate the metric dimension of the random graph for a wide range
of probabilities
Bounds and extremal graphs for total dominating identifying codes
An identifying code of a graph is a dominating set of such that
any two distinct vertices of have distinct closed neighbourhoods within
. The smallest size of an identifying code of is denoted
. When every vertex of also has a neighbour in ,
it is said to be a total dominating identifying code of , and the smallest
size of a total dominating identifying code of is denoted by
.
Extending similar characterizations for identifying codes from the
literature, we characterize those graphs of order with
(the only such connected graph is ) and
(such graphs either satisfy
or are built from certain such graphs by adding a
set of universal vertices, to each of which a private leaf is attached).
Then, using bounds from the literature, we remark that any (open and closed)
twin-free tree of order has a total dominating identifying code of size at
most . This bound is tight, and we characterize the trees
reaching it. Moreover, by a new proof, we show that this bound actually holds
for the larger class of all twin-free graphs of girth at least 5. The cycle
also attains this bound. We also provide a generalized bound for all
graphs of girth at least 5 (possibly with twins).
Finally, we relate to the related parameter
as well as the location-domination number of and
its variants, providing bounds that are either tight or almost tight
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 2 ln(|V|)+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 |V|^a with a in {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
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