2020/01/24 by Lucas Mol, Mol, Lucas, Matthew J. H. Murphy +3 · 1 citation
Mathematics · #05C05 #05C12 #05C69 #05C75 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C05 #msc:05C12 #msc:05C69 #msc:05C75
paper · pdf · doi:10.48550/arxiv.2001.09168
27 pages, 14 Figures
arxiv created 2020/01/24 · arxiv updated 2020/01/28
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 β(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 τ(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 β(G) = τ(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 β(H)=τ(G), then H is called a threshold graph of G. The threshold dimension of a graph is expressed in terms of a minimum number of strong products of paths that admits a certain type of embedding of the graph. 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.