On the multiset dimension of rectangular king grids

A set S of vertices of a connected graph G is called a multiset resolving set if no two vertices of G have the same multiset of distances to S, and the smallest size of such a set is the multiset dimension dimm(G). For squares dimm(Pn ⊠ Pn) = 4 when n ≥ 5 (Hakanen and Yero, Allikvere), and for strips of fixed height dimm(Ph ⊠ Pn) grows linearly with n. We show that dimm(Ph ⊠ Pn) = 4 for h ≥ 5 and h ≤ n ≤ ⌊3h/2⌋ − 1, with explicit resolving sets for odd and even h, and that dimm(Ph ⊠ Pn) ≥ 4 for all n ≥ h ≥ 3. Counting columns gives dimm(Ph ⊠ Pn) ≥ ⌈n/(h − 1)⌉. A finite search shows that for 3 ≤ h ≤ 28 no grid Ph ⊠ Pn with n ≥ ⌊3h/2⌋ has a multiset resolving set of size four. We conjecture that for every h ≥ 5 four vertices suffice exactly when h ≤ n ≤ ⌊3h/2⌋ − 1.