Vertaisarvioitu alkuperäisartikkeli tai data-artikkeli tieteellisessä aikakauslehdessä (A1)
On Vertices Contained in All or in No Metric Basis
Julkaisun tekijät: Hakanen Anni, Junnila Ville, Laihonen Tero, Yero Ismael G.
Kustantaja: Elsevier BV
Julkaisuvuosi: 2021
Journal: Discrete Applied Mathematics
eISSN: 1872-6771
DOI: http://dx.doi.org/10.1016/j.dam.2021.12.004
Verkko-osoite: https://www.sciencedirect.com/science/article/pii/S0166218X21004820?via%3Dihub
Rinnakkaistallenteen osoite: https://research.utu.fi/converis/portal/detail/Publication/68601224
A set R ⊆ V (G) is a resolving set of a graph G if for all distinct vertices v, u ∈ V (G) there exists an element r ∈ R such that d(r, v) ̸ = d(r, u). The metric dimension dim(G) of the graph G is the cardinality of a smallest resolving set of G. A resolving set with cardinality dim(G) is called a metric basis of G. We consider vertices that are in all metric bases, and we call them basis forced vertices. We give several structural properties of sparse and dense graphs where basis forced vertices are present. In particular, we give bounds for the maximum number of edges in a graph containing basis forced vertices. Our bound is optimal whenever the number of basis forced vertices is even. Moreover, we provide a method of constructing fairly sparse graphs with basis forced vertices. We also study vertices which are in no metric basis in connection to cut-vertices and pendants. Furthermore, we show that deciding whether a vertex is in all metric bases is co-NP-hard, and deciding whether a vertex is in no metric basis is NP-hard.
Ladattava julkaisu This is an electronic reprint of the original article. | ||
Ladattava julkaisu This is an electronic reprint of the original article. |