Partition Dimension and Strong Metric Dimension of Chain Cycle

Abstract

Let G be a connected graph with vertex set V(G) and edge set E(G). For an ordered k-partition ӏӏ = {Q₁, ... ,Q_k} of V(G), the representation of a vertex v ∈ V(G) with respect to ӏӏ is the k-vectors r(v|ӏӏ) = (d(v,Q₁), ... , d(v,Q_k)), where d(v,Q_i) is the distance between v and Q_i. The partition ӏӏ is a resolving partition if r(u|ӏӏ) ≠ r(v|ӏӏ), for each pair of distinct vertices u, v ∈ V(G). The minimum k for which there is a resolving k-partition of V(G) is the partition dimension of G. A vertex w ∈ V(G) strongly resolves two distinct vertices u, v∈V(G) if u belongs to a shortest v - w path or v belongs to a shortest u - w path. An ordered set W = {w₁, ... ,w_t} ⊆ V(G) is a strong resolving set for G if for every two distinct vertices u and v of G there exists a vertex w ∈ W which strongly resolves u and v. A strong metric basis of G is a strong resolving set of minimal cardinality. The cardinality of a strong metric basis is called strong metric dimension of G. In this paper, we determine the partition dimension and strong metric dimension of a chain cycle constructed by even cycles and a chain cycle constructed by odd cycles.

Key words and phrases. Partition dimension, strong metric dimension, chain cycle.

2000 Mathematics Subject Classification. 05C12