Resolving dominating partitions in graphs

Abstract

A partition Π={S1,…,Sk} of the vertex set of a connected graph G is called a resolving partition of G if for every pair of vertices u and v, d(u,Sj)≠d(v,Sj), for some part Sj. The partition dimensionβp(G) is the minimum cardinality of a resolving partition of G. A resolving partition Π is called resolving dominating if for every vertex v of G, d(v,Sj)=1, for some part Sj of Π. The dominating partition dimensionηp(G) is the minimum cardinality of a resolving dominating partition of G.In this paper we show, among other results, that βp(G)≤ηp(G)≤βp(G)+1. We also characterize all connected graphs of order n≥7 satisfying any of the following conditions: ηp(G)=n, ηp(G)=n−1, ηp(G)=n−2 and βp(G)=n−2. Finally, we present some tight Nordhaus–Gaddum bounds for both the partition dimension βp(G) and the dominating partition dimension ηp(G).