4,239 research outputs found
Identifying and locating-dominating codes on chains and cycles
AbstractConsider a connected undirected graph G=(V,E), a subset of vertices C⊆V, and an integer r≥1; for any vertex v∈V, let Br(v) denote the ball of radius r centered at v, i.e., the set of all vertices within distance r from v. If for all vertices v∈V (respectively, v∈V ⧹C), the sets Br(v)∩C are all nonempty and different, then we call C an r-identifying code (respectively, an r-locating-dominating code). We study the smallest cardinalities or densities of these codes in chains (finite or infinite) and cycles
Identifying codes and locating–dominating sets on paths and cycles
AbstractLet G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define Nr[x]={y∈V:d(x,y)≤r} and Dr(x)=Nr[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V∖D, respectively), Dr(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72–82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating–dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969–987]
On two variations of identifying codes
Identifying codes have been introduced in 1998 to model fault-detection in
multiprocessor systems. In this paper, we introduce two variations of
identifying codes: weak codes and light codes. They correspond to
fault-detection by successive rounds. We give exact bounds for those two
definitions for the family of cycles
On global location-domination in graphs
A dominating set of a graph is called locating-dominating, LD-set for
short, if every vertex not in is uniquely determined by the set of
neighbors of belonging to . Locating-dominating sets of minimum
cardinality are called -codes and the cardinality of an LD-code is the
location-domination number . An LD-set of a graph is global
if it is an LD-set of both and its complement . The global
location-domination number is the minimum cardinality of a
global LD-set of . In this work, we give some relations between
locating-dominating sets and the location-domination number in a graph and its
complement.Comment: 15 pages: 2 tables; 8 figures; 20 reference
Solving Two Conjectures regarding Codes for Location in Circulant Graphs
Identifying and locating-dominating codes have been widely studied in
circulant graphs of type , which can also be viewed as
power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered
identification and location-domination in the circulant graphs . They
showed that the smallest cardinality of a locating-dominating code in
is at least and at most
for all . Moreover, they proved that the lower bound is strict when
and conjectured that the lower bound can be
increased by one for other . In this paper, we prove their conjecture.
Similarly, they showed that the smallest cardinality of an identifying code in
is at least and at most for all . Furthermore, they proved that the lower bound is
attained for most of the lengths and conjectured that in the rest of the
cases the lower bound can improved by one. This conjecture is also proved in
the paper. The proofs of the conjectures are based on a novel approach which,
instead of making use of the local properties of the graphs as is usual to
identification and location-domination, also manages to take advantage of the
global properties of the codes and the underlying graphs
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
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
Identifying codes of corona product graphs
For a vertex of a graph , let be the set of with all of
its neighbors in . A set of vertices is an {\em identifying code} of
if the sets are nonempty and distinct for all vertices . If
admits an identifying code, we say that is identifiable and denote by
the minimum cardinality of an identifying code of . In this
paper, we study the identifying code of the corona product of graphs
and . We first give a necessary and sufficient condition for the
identifiable corona product , and then express in terms of and the (total) domination number of .
Finally, we compute for some special graphs
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