Dorota Kuziak

On the strong metric dimension of Cartesian and direct products of graphs

Abstract Let G be a connected graph. A vertex w strongly resolves a pair u , v of vertices of G if there exists some shortest u − w path containing v or some shortest v − w path containing u . A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W . The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G . It is known that the problem of computing the strong metric dimension of a graph is NP-hard. In this paper we obtain closed formulae for the strong metric dimension of several families of Cartesian products of graphs and direct products of graphs.

On the strong metric dimension of the strong products of graphs

Abstract Let G be a connected graph. A vertex w ∈ V.G/ strongly resolves two vertices u,v ∈ V.G/ if there exists some shortest u-w path containing v or some shortest v-w path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. It is well known that the problem of computing this invariant is NP-hard. In this paper we study the problem of finding exact values or sharp bounds for the strong metric dimension of strong product graphs and express these in terms of invariants of the factor graphs.

The partition dimension of strong product graphs and Cartesian product graphs

Abstract Let G = ( V , E ) be a connected graph. The distance between two vertices u , v ∈ V , denoted by d ( u , v ) , is the length of a shortest u , v -path in G . The distance between a vertex v ∈ V and a subset P ⊂ V is defined as min { d ( v , x ) : x ∈ P } , and it is denoted by d ( v , P ) . An ordered partition { P 1 , P 2 , … , P t } of vertices of a graph G , is a resolving partition of G , if all the distance vectors ( d ( v , P 1 ) , d ( v , P 2 ) , … , d ( v , P t ) ) are different. The partition dimension of G is the minimum number of sets in any resolving partition of G . In this article we study the partition dimension of strong product graphs and Cartesian product graphs. Specifically, we prove that the partition dimension of the strong product of graphs is bounded below by four and above by the product of the partition dimensions of the factor graphs. Also, we give the exact value of the partition dimension of strong product graphs when one factor is a complete graph and the other one is a path or a cycle. For the case of Cartesian product graphs, we show that its partition dimension is less than or equal to the sum of the partition dimensions of the factor graphs minus one. Moreover, we obtain an upper bound on the partition dimension of Cartesian product graphs, when one factor is a complete graph.

Strong metric dimension of rooted product graphs

Let G be a connected graph. A vertex w strongly resolves two different vertices of G if there exists a shortest path, which contains the vertex v or a shortest path, which contains the vertex u. A set W of vertices is a strong metric generator for G if every pair of different vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong metric generator for G is called the strong metric dimension of G. It is known that the problem of computing this invariant is NP-hard. According to that fact, in this paper we study the problem of computing exact values or sharp bounds for the strong metric dimension of the rooted product of graphs and express these in terms of invariants of the factor graphs.

Closed Formulae for the Strong Metric Dimension of Lexicographic Product Graphs

Abstract Given a connected graph G, a vertex w ∈ V (G) strongly resolves two vertices u, v ∈ V (G) if there exists some shortest u − w path containing v or some shortest v − w path containing u. A set S of vertices is a strong metric generator for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong metric generator for G is called the strong metric dimension of G. In this paper we obtain several relationships between the strong metric dimension of the lexicographic product of graphs and the strong metric dimension of its factor graphs.

Computing the Metric Dimension of a Graph from Primary Subgraphs

Abstract Let G be a connected graph. Given an ordered set W = {w1, . . . , wk} ⊆ V (G) and a vertex u ∈ V (G), the representation of u with respect to W is the ordered k-tuple (d(u, w1), d(u, w2), . . . , d(u, wk)), where d(u, wi) denotes the distance between u and wi. The set W is a metric generator for G if every two different vertices of G have distinct representations. A minimum cardinality metric generator is called a metric basis of G and its cardinality is called the metric dimension of G. It is well known that the problem of finding the metric dimension of a graph is NP-hard. In this paper we obtain closed formulae for the metric dimension of graphs with cut vertices. The main results are applied to specific constructions including rooted product graphs, corona product graphs, block graphs and chains of graphs.

Erratum to “On the strong metric dimension of the strong products of graphs”

Abstract The original version of the article was published in Open Mathematics (formerly Central European Journal of Mathematics) 13 (2015) 64–74. Unfortunately, the original version of this article contains a mistake: in Lemma 2.17 appears that for any C1-graph G and any graph H, β (G ⊠ H) ≤ β (G)(β(H)+1), while should be β (G ⊠ H) ≤ β(H) (β(G)+1). In this erratum we correct the lemma, its proof and some of its consequences.

Resolvability and Strong Resolvability in the Direct Product of Graphs

Given a connected graph G, a vertex

The security number of strong grid-like graphs

{w \in V(G)}