5 research outputs found
Location-domination and matching in cubic graphs
A dominating set of a graph is a set of vertices of such that
every vertex outside is adjacent to a vertex in . A locating-dominating
set of 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 . Garijo, Gonzalez and Marquez
[Applied Math. Computation 249 (2014), 487--501] posed the conjecture that for
sufficiently large, the maximum value of the location-domination number of
a twin-free, connected graph on vertices is equal to . We propose the related (stronger) conjecture that if is a
twin-free graph of order without isolated vertices, then . We prove the conjecture for cubic graphs. We rely heavily on
proof techniques from matching theory to prove our result.Comment: 16 pages; 4 figure
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
Domination and location in twin-free digraphs
A dominating set in a digraph is a set of vertices such that every vertex
is either in or has an in-neighbour in . A dominating set of a
digraph is locating-dominating if every vertex not in has a unique set of
in-neighbours within . The location-domination number of a
digraph is the smallest size of a locating-dominating set of . We
investigate upper bounds on in terms of the order of . We
characterize those digraphs with location-domination number equal to the order
or the order minus one. Such digraphs always have many twins: vertices with the
same (open or closed) in-neighbourhoods. Thus, we investigate the value of
in the absence of twins and give a general method for
constructing small locating-dominating sets by the means of special dominating
sets. In this way, we show that for every twin-free digraph of order ,
holds, and there exist twin-free digraphs
with . If moreover is a tournament or is
acyclic, the bound is improved to ,
which is tight in both cases
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
Progress towards the two-thirds conjecture on locating-total dominating sets
We study upper bounds on the size of optimum locating-total dominating sets
in graphs. A set of vertices of a graph is a locating-total dominating
set if every vertex of has a neighbor in , and if any two vertices
outside have distinct neighborhoods within . The smallest size of such a
set is denoted by . It has been conjectured that
holds for every twin-free graph of order
without isolated vertices. We prove that the conjecture holds for
cobipartite graphs, split graphs, block graphs, subcubic graphs and outerplanar
graphs