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ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

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Discussiones Mathematicae Graph Theory

CONNECTED PARTITION DIMENSIONS OF GRAPHS

Varaporn Saenpholphat and Ping Zhang

Department of Mathematics and Statistics
Western Michigan University
Kalamozoo, MI 49008, USA

Abstract

For a vertex v of a connected graph G and a subset S of V(G), the distance between v and S is d(v,S) = min{d(v,x)|x ∈ S}. For an ordered k-partition Π={S1,S2,…,Sk} of V(G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v,S1), d(v,S2),…, d(v,Sk)). The k-partition Π is a resolving partition if the k-vectors r(v|Π), v ∈ V(G), are distinct. The minimum k for which there is a resolving k-partition of V(G) is the partition dimension pd(G) of G. A resolving partition Π = {S1,S2,…,Sk} of V(G) is connected if each subgraph ⟨Si⟩ induced by Si (1 ≤ i ≤ k) is connected in G. The minimum k for which there is a connected resolving k-partition of V(G) is the connected partition dimension cpd(G) of G. Thus 2 ≤ pd (G) ≤cpd(G) ≤ n for every connected graph G of order n≥2. The connected partition dimensions of several classes of well-known graphs are determined. It is shown that for every pair a, b of integers with 3≤a≤b≤2a−1, there is a connected graph G having pd(G) = a and cpd(G) = b. Connected graphs of order n ≥ 3 having connected partition dimension 2, n, or n−1 are characterized.

Keywords: distance, resolving partition, connected resolving partition.

2000 Mathematics Subject Classification: 05C12.

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