The threshold dimension of a graph

Let G be a graph, and let u, v, and w be vertices of G. If the distance between u and w does not equal the distance between v and w, then w is said to resolve u and v. The metric dimension of G, denoted beta(G), is the cardinality of a smallest set W of vertices such that every pair of vertices of G is resolved by some vertex of W. The threshold dimension of a graph G, denoted tau(G), is the minimum metric dimension among all graphs H having G as a spanning subgraph. In other words, the threshold dimension of G is the minimum metric dimension among all graphs obtained from G by adding edges. If beta(G) = tau(G), then G is said to be irreducible; otherwise, we say that G is reducible. If H is a graph having G as a spanning subgraph and such that beta(H) = tau(G), then H is called a threshold graph of G. The first main part of the paper has a geometric flavour, and gives an expression for the threshold dimension of a graph in terms of a minimum number of strong products of paths (each of sufficiently large order) that admits a certain type of embedding of the graph. This result is used to show that there are trees of arbitrarily large metric dimension having threshold dimension 2. The second main part of the paper focuses on the threshold dimension of trees. A sharp upper bound for the threshold dimension of trees is established. It is also shown that the irreducible trees are precisely those of metric dimension at most 2. Moreover, if T is a tree with metric dimension 3 or 4, then T has threshold dimension 2. It is shown, in these two cases, that a threshold graph for T can be obtained by adding exactly one or two edges to T, respectively. However, these results do not extend to trees with metric dimension 5, i.e., there are trees of metric dimension 5 with threshold dimension exceeding 2. (C) 2020 Elsevier B.V. All rights reserved.