Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vertex and edge metric dimensions of cacti   In a graph G; a vertex (resp. an edge) metric generator is a set of vertices
S such that any pair of vertices (resp. edges) from G is distinguished by at
least one vertex from S: The cardinality of a smallest vertex (resp. edge)
metric generator is the vertex (resp. edge) metric dimension of G: In [19] we
determined the vertex (resp. edge) metric dimension of unicyclic graphs and
that it takes its value from two consecutive integers. Therein, several cycle
configurations were introduced and the vertex (resp. edge) metric dimension
takes the greater of the two consecutive values only if any of these
configurations is present in the graph. In this paper we extend the result to
cactus graphs i.e. graphs in which all cycles are pairwise edge disjoint. We do
so by defining a unicyclic subgraph of G for every cycle of G and applying the
already introduced approach for unicyclic graphs which involves the
configurations. The obtained results enable us to prove the cycle rank
conjecture for cacti. They also yield a simple upper bound on metric dimensions
of cactus graphs and we conclude the paper by conjecturing that the same upper
bound holds in general.

17
The categories of this paper are: q-fin
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
13
The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org