Distance-based vertex identification in graphs: The outer multiset dimension

The characterisation of vertices in a network, in relation to other peers, has been used as a primitive in many computational procedures, such as node localisation and (de-)anonymisation. This article focuses on a characterisation type known as the multiset metric representation. Formally, given a graph G and a subset of vertices S={w1,…,wt}⊆V(G), the multiset representationof a vertex u ∈ V(G) with respect to S is the multiset m(u|S)={|dG(u,w1),…,dG(u,wt)|}. A subset of vertices S such that m(u|S)=m(v|S)⇔u=v for every u, v ∈ V(G)∖S is said to be a multiset resolving set, and the cardinality of the smallest such set is the outer multiset dimension. We study the general behaviour of the outer multiset dimension, and determine its exact value for several graph families. We also show that computing the outer multiset dimension of arbitrary graphs is NP-hard, and provide methods for efficiently handling particular cases.