7,402 research outputs found
Resolving sets for breaking symmetries of graphs
This paper deals with the maximum value of the difference between the
determining number and the metric dimension of a graph as a function of its
order. Our technique requires to use locating-dominating sets, and perform an
independent study on other functions related to these sets. Thus, we obtain
lower and upper bounds on all these functions by means of very diverse tools.
Among them are some adequate constructions of graphs, a variant of a classical
result in graph domination and a polynomial time algorithm that produces both
distinguishing sets and determining sets. Further, we consider specific
families of graphs where the restrictions of these functions can be computed.
To this end, we utilize two well-known objects in graph theory: -dominating
sets and matchings.Comment: 24 pages, 12 figure
New results on metric-locating-dominating sets of graphs
A dominating set of a graph is a metric-locating-dominating set if each
vertex of the graph is uniquely distinguished by its distances from the
elements of , and the minimum cardinality of such a set is called the
metric-location-domination number. In this paper, we undertake a study that, in
general graphs and specific families, relates metric-locating-dominating sets
to other special sets: resolving sets, dominating sets, locating-dominating
sets and doubly resolving sets. We first characterize classes of trees
according to certain relationships between their metric-location-domination
number and their metric dimension and domination number. Then, we show
different methods to transform metric-locating-dominating sets into
locating-dominating sets and doubly resolving sets. Our methods produce new
bounds on the minimum cardinalities of all those sets, some of them involving
parameters that have not been related so far.Comment: 13 pages, 3 figure
New results on metric-locating-dominating sets of graphs
A dominating set S of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distanc es from the elements of S , and the minimum cardinality of such a set is called the metri c-location- domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominatin g sets to other special sets: resolving sets, dominating sets, locating-dominating set s and doubly resolving sets. We first characterize classes of trees according to cer tain relationships between their metric-location-domination number and thei r metric dimension and domination number. Then, we show different methods to tran sform metric- locating-dominating sets into locating-dominating sets a nd doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them involving parameters that have not been related so farPostprint (published version
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
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
Locating-dominating sets in twin-free graphs
A locating-dominating set of a graph is a 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-domination number of , denoted
, is the minimum cardinality of a locating-dominating set in .
It is conjectured [D. Garijo, A. Gonz\'alez and A. M\'arquez. The difference
between the metric dimension and the determining number of a graph. Applied
Mathematics and Computation 249 (2014), 487--501] that if is a twin-free
graph of order without isolated vertices, then . We prove the general bound ,
slightly improving over the bound of Garijo et
al. We then provide constructions of graphs reaching the bound,
showing that if the conjecture is true, the family of extremal graphs is a very
rich one. Moreover, we characterize the trees that are extremal for this
bound. We finally prove the conjecture for split graphs and co-bipartite
graphs.Comment: 11 pages; 4 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
Locating and Total Dominating Sets of Direct Products of Complete Graphs
A set S of vertices in a graph G = (V,E) is a metric-locating-total dominating set of G if every vertex of V is adjacent to a vertex in S and for every u ≠ v in V there is a vertex x in S such that d(u,x) ≠ d(v,x). The metric-location-total domination number \gamma^M_t(G) of G is the minimum cardinality of a metric-locating-total dominating set in G. For graphs G and H, the direct product G × H is the graph with vertex set V(G) × V(H) where two vertices (x,y) and (v,w) are adjacent if and only if xv in E(G) and yw in E(H). In this paper, we determine the lower bound of the metric-location-total domination number of the direct products of complete graphs. We also determine some exact values for some direct products of two complete graphs
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
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