5 research outputs found
Characterization of Randomly k-Dimensional Graphs
For an ordered set of vertices and a vertex in a
connected graph , the ordered -vector
is called the (metric) representation
of with respect to , where is the distance between the vertices
and . The set is called a resolving set for if distinct vertices
of have distinct representations with respect to . A minimum resolving
set for is a basis of and its cardinality is the metric dimension of
. The resolving number of a connected graph is the minimum , such
that every -set of vertices of is a resolving set. A connected graph
is called randomly -dimensional if each -set of vertices of is a
basis. In this paper, along with some properties of randomly -dimensional
graphs, we prove that a connected graph with at least two vertices is
randomly -dimensional if and only if is complete graph or an
odd cycle.Comment: 12 pages, 3 figure
The resolving number of a graph
We study a graph parameter related to resolving sets and metric dimension,
namely the resolving number, introduced by Chartrand, Poisson and Zhang. First,
we establish an important difference between the two parameters: while
computing the metric dimension of an arbitrary graph is known to be NP-hard, we
show that the resolving number can be computed in polynomial time. We then
relate the resolving number to classical graph parameters: diameter, girth,
clique number, order and maximum degree. With these relations in hand, we
characterize the graphs with resolving number 3 extending other studies that
provide characterizations for smaller resolving number.Comment: 13 pages, 3 figure
On the metric dimension, the upper dimension and the resolving number of graphs
This paper deals with three resolving parameters: the metric dimension, the upper dimension and the resolving number. We first answer a question raised by Chartrand and Zhang asking for a characterization of the graphs with equal metric dimension and resolving number. We also solve in the affirmative a conjecture posed by Chartrand, Poisson and Zhang about the realization of the metric dimension and the upper dimension. Finally, we prove that no integer a≥4a≥4 is realizable as the resolving number of an infinite family of graphs