On the outer multiset dimensions of graphs

Abstract An outer multiset resolving set uniquely identifies the vertices of a given graph. A subset $$W \subseteq V(G)$$ W ⊆ V ( G ) is an outer multiset resolving set if every two distinct vertices in $$V(G){\setminus } W$$ V ( G ) \ W have distinct multisets of distances to W . The smallest such set is an outer multiset basis, and its cardinality is the outer multiset dimension of G . This notion is closely related to the multiset resolving set, which distinguishes all vertices of G , while outer multisets only distinguish those outside the set. In this paper, for integers $$d,c \ge 1$$ d , c ≥ 1 , we construct an infinite family of graphs with diameter d admitting a multiset resolving set $$W = \{w_1, \ldots , w_c\}$$ W = { w 1 , … , w c } of maximum order $$n = {d + c - 1 \atopwithdelims ()d - 1} + c$$ n = d + c - 1 d - 1 + c . Besides, we are interested in finding the outer multiset dimension of an infinite family of graphs. Since the outer multiset dimension is known for cycles, we took the circulant graphs $$C_n(1,2)$$ C n ( 1 , 2 ) of order n as a natural generalization of the cycles.