Given a connected simple graph G = (V, E), and a positive integer k, a set S ⊆ V is said to be a k-metric generator for G if and only if for any pair of different vertices u, v ∈ V , there exist at least k vertices w 1 , w 2 , ...,is the length of a shortest path between x and y. A k-metric generator of minimum cardinality in G is called a k-metric basis and its cardinality, the k-metric dimension of G. In this article we study the k-metric dimension of corona product graphs G ⊙ H, where G is a graph of order n and H is a family of n non-trivial graphs. Specifically, we give some necessary and sufficient conditions for the existence of a k-metric basis in a connected corona graph. Moreover, we obtain tight bounds and closed formulae for the k-metric dimension of connected corona graphs.

Given a connected graph G=(V(G),E(G)), a set S⊆V(G) is said to be a k-metric generator for G if any pair of different vertices in V(G) is distinguished by at least k elements of S. A metric generator of minimum cardinality among all k-metric generators is called a k-metric basis and its cardinality is the k-metric dimension of G. We initially present a linear programming problem that describes the problem of finding the k-metric dimension and a k-metric basis of a graph G. Then we conducted a study on the k-metric dimension of a unicyclic graph.

Let $(X,d)$ be a metric space. A set $S\subseteq X$ is said to be a $k$-metric generator for $X$ if and only if for any pair of different points $u,v\in X$, there exist at least $k$ points $w_1,w_2, \ldots w_k\in S$ such that $d(u,w_i)\ne d(v,w_i),\; \textrm{for all}\; i\in \{1, \ldots k\}.$ Let $\mathcal{R}_k(X)$ be the set of metric generators for $X$. The $k$-metric dimension $\dim _k(X)$ of $(X,d)$ is defined as $$\begin{equation*}\dim_k(X)=\inf\{|S|:\, S\in \mathcal{R}_k(X)\}.\end{equation*}$$Here, we discuss the $k$-metric dimension of $(V,d_t)$, where $V$ is the set of vertices of a simple graph $G$ and the metric $d_t:V\times V\rightarrow \mathbb{N}\cup \{0\}$ is defined by $d_t(x,y)=\min \{d(x,y),t\}$ from the geodesic distance $d$ in $G$ and a positive integer $t$. The case $t\ge D(G)$, where $D(G)$ denotes the diameter of $G$, corresponds to the original theory of $k$-metric dimension, and the case $t=2$ corresponds to the theory of $k$-adjacency dimension. Furthermore, this approach allows us to extend the theory of $k$-metric dimension to the general case of non-necessarily connected graphs. Finally, we analyse the computational complexity of determining the $k$-metric dimension of $(V,d_t)$ for the metric $d_t$.

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The multiperiod vehicle routing problem (MPVRP) is an extension of the vehicle routing problem in which customer demands have to be delivered in one of several consecutive time periods, for example, the days of a week. We introduce and explore a variant of the MPVRP in which the carrier offers a price discount in exchange for delivery flexibility. The carrier's goal is to minimize total costs, which consist of the distribution costs and the discounts paid. A biased-randomized iterated local search algorithm is proposed for its solution. The two-stage algorithm first quickly generates a number of promising customer-to-period assignments, and then intensively explores a subset of these assignments. An extensive computational study demonstrates the efficacy of the proposed algorithm and highlights the benefit of pricing for delivery flexibility in different settings.

A generator of a metric space is a set S of points in the space with the property that every point of the space is uniquely determined by its distances from the elements of S. Given a simple graph G = (V, E), we define the distance function dG,2 : V × V → N ∪ {0}, as dG,2(x, y) = min{dG(x, y), 2}, where dG(x, y) is the length of a shortest path between x and y and N is the set of positive integers. Then (V, dG,2) is a metric space. We say that a set S ⊆ V is a k-adjacency generator for G if for every two vertices x, y ∈ V , there exist at least k vertices w1, w2, ..., w k ∈ S such that dG,2(x, wi) = dG,2(y, wi), for every i ∈ {1, ..., k}.A minimum cardinality k-adjacency generator is called a k-adjacency basis of G and its cardinality, the k-adjacency dimension of G.In this article we study the problem of finding the k-adjacency dimension of a graph. We give some necessary and sufficient conditions for the existence of a k-adjacency basis of an arbitrary graph G and we obtain general results on the k-adjacency dimension, including general bounds and closed formulae for some families of graphs. In particular, we obtain closed formulae for the k-adjacency dimension of join graphs G + H in terms of the k-adjacency dimension of G and H. These results concern the k-metric dimension, as join graphs have diameter two. As we can expect, the obtained results will become important tools for the study of the k-metric dimension of lexicographic product graphs and corona product graphs. Moreover, several results obtained in this paper need not be restricted to the metric dG,2, they can be expressed in a more general setting, for instance, by using the metric dG,t(x, y) = min{dG(x, y), t} for t ∈ N.

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