Some binary products and integer linear programming for computing $k$-metric dimension of graphs

Abstract

If S = {v 1 , . . . , v ℓ } is an ordered subset of vertices of a connected graph G and v is a vertex of G, then the vector rIf for any pair of different vertices u, v ∈ V (G), the vectors r(v|S) and r(u|S) differ in at least k positions, then S is a k-metric generator for G. A smallest k-metric generator for G is a k-metric basis for G, its cardinality being the k-metric dimension of G. A sharp upper bound and a closed formulae for the k-metric dimension of the hierarchical product of graphs is proved. Also, sharp lower bounds for the k-metric dimension of the splice and link products of graphs are presented. An integer linear programming model for computing the k-metric dimension and a k-metric basis of a given graph is proposed. These results are applied to bound or to compute the k-metric dimension of some classes of graphs that are of interest in mathematical chemistry.

Metric dimension related parameters in graphs: A survey on combinatorial, computational and applied results

“…Following with the last notes from the previous subsection, we now center our attention into the k-metric dimension of product graphs. The main contributions to this topic are centered into three products: lexicographic ( [61,64]), hierarchical ( [109]) and corona ( [63]) products, although some sporadic results have appeared for some other products. Namely, still several contributions on the k-metric dimension of product graphs could enrich this theory.…”

“…Finally, to end this subsection, we mention that the k-metric dimension of the hierarchical product of graphs was studied in [109] together with other less common graph operations called splice and link products. The most remarkable aspect of this work is the application of some integer linear programming formulation while computing the k-metric dimension of special cases of the hierarchical product graph dim k (G(U )⊓H) (with respect to a set U ⊆ V (G)), aimed to show the tightness of the main bound of the article, which is next stated.…”

“…The most remarkable aspect of this work is the application of some integer linear programming formulation while computing the k-metric dimension of special cases of the hierarchical product graph dim k (G(U )⊓H) (with respect to a set U ⊆ V (G)), aimed to show the tightness of the main bound of the article, which is next stated. For definitions of G(U ) ⊓ H, see [109] precisely.…”

Resolving sets tolerant to failures in three-dimensional grids

“…Hence, a set S remains resolving even though k vertices fail if and only if S is a (k + 1)-resolving set. The k-metric dimension and k-resolving sets of a graph and, concretely, of some product graphs have been studied in [2,[9][10][11][12]18,22,28]. In the survey [20], an extensive summary of known results and applications of the k-metric dimension is given.…”

Resolving sets tolerant to failures in three-dimensional grids

“…Hence, a set S remains resolving even though k vertices fail if and only if S is a (k +1)-resolving set. The k-metric dimension and k-resolving sets of a graph and, concretely, of some product graphs have been studied in [2,9,10,11,12,18,22,28]. In the survey [20], an extensive summary of known results and applications of the k-metric dimension is given.…”