The solid-metric dimension

On the metric dimensions for sets of vertices

“…On the Metric Dimensions for Sets of Vertices 5 Theorem 3 will be very useful throughout the article. This theorem also implies the corresponding result for = 1 in [7,Thm 2.2]. A somewhat similar result holds for { }-resolving sets as stated in the following lemma.…”

“…In this section, we prove a lower bound on the -solid-metric dimension of a graph and characterise the graphs attaining that bound. The lower bound β s 1 (G) ≥ 2 on the 1-solidmetric dimension of a graph was shown in [7]. The following theorem generalises this lower bound for -solid-metric dimensions where ≥ 2.…”

“…Thus, no such graph has forced vertices for a {1}-resolving set. In [7], the forced vertices of 1-solid-resolving sets were fully characterised. In this section, we prove characterisations for -solid-and { }-resolving sets for all .…”

“…The set S ⊆ V (G) is an -solid-resolving set of G, if for all distinct nonempty sets X, Y ⊆ V (G) such that |X| ≤ we have D S (X) = D S (Y ). When = 1, the previous definition is exactly the same as the definition of a solid-resolving set in [7]. The set 9 } is a 2-solidresolving set of H. We can distinguish the sets U and W from each other using S 2 since D S 2 (U ) = (2, 1, 2, 1, 1, 1, 1) and D S 2 (W ) = (2, 1, 2, 1, 0, 1, 1).…”

“…In [7], it was shown that a 1-solid-resolving set of G is a doubly resolving set of G. According to Theorem 5 any { }-resolving set, where ≥ 2, and -solidresolving set is a 1-solid-resolving set. The following result is now immediate.…”