As a generalization of the concept metric basis, this article introduces notion $k$-metric basis in graphs. Given connected graph $G=(V,E)$, set $S\subseteq V$ is said to be generator for $G$ if elements any pair different vertices are distinguished by at least $k$ $S$, i.e., two $u,v\in V$, there exist $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ every $i\in \{1,...,k\}$. A minimum cardinality called and its dimension $G$. dimensional largest integer exists We give necessary...

Abstract The multiperiod vehicle routing problem (MPVRP) is an extension of the in which customer demands have to be delivered one several consecutive time periods, for example, days a week. We introduce and explore variant MPVRP carrier offers price discount exchange delivery flexibility. carrier's goal minimize total costs, consist distribution costs discounts paid. A biased‐randomized iterated local search algorithm proposed its solution. two‐stage first quickly generates number promising...

Abstract In the context of a supply chain for animal‐feed industry, this paper focuses on optimizing replenishment strategies silos in multiple farms. Assuming that is essentially value chain, our work aims at narrowing chasm and putting analytics into practice by identifying quantifying improvements specific stages an chain. Motivated real‐life case, analyses rich multi‐period inventory routing problem with homogeneous fleet, stochastic demands, maximum route length. After describing...

Abstract Let $(X,d)$ be a metric space. A set $S\subseteq X$ is said to $k$-metric generator for $X$ if and only any pair of different points $u,v\in X$, there exist at least $k$ $w_1,w_2, \ldots w_k\in S$ such that $d(u,w_i)\ne d(v,w_i),\; \textrm{for all}\; i\in \{1, k\}.$ $\mathcal{R}_k(X)$ the generators $X$. The dimension $\dim _k(X)$ defined as $$\begin{equation*}\dim_k(X)=\inf\{|S|:\, S\in \mathcal{R}_k(X)\}.\end{equation*}$$Here, we discuss $(V,d_t)$, where $V$ vertices simple graph...

In this article we study the problem of finding k-adjacency dimension a graph. We give some necessary and sufficient conditions for existence basis an arbitrary graph G obtain general results on dimension, including bounds closed formulae families graphs.

The concepts of unmanned aerial vehicles and self-driving are gaining relevance inside the smart city environment. This type might use ultra-reliable telecommunication systems, Internet-based technologies, navigation satellite services to decide about routes they must follow efficiently accomplish their mission reach destinations in due time. When working teams vehicles, there is a need coordinate routing operations. some unexpected events occur (e.g., after traffic accident, natural...

Abstract In supermarkets, perishable products need to be sold consumers before a given deadline, after which their monetary value is significantly diminished or even completely lost. the case of valuable that should not wasted, following operational decision needs made as this deadline approaches: best way reallocate from stores with surplus inventories unsatisfied demand? This question results in an optimization problem goal minimize total transport cost plus opportunity associated...

In this paper we obtain closed formulae for several parameters of generalized Sierpiński graphs S(G, t) in terms the base graph G.In particular, focus on chromatic, vertex cover, clique and domination numbers.

A set D of vertices a graph G is double dominating if |N[v]∩D|≥2 for every v∈V(G), where N[v] represents the closed neighbourhood v. The domination number minimum cardinality among all sets G. In this article, we show that and H are graphs with no isolated vertex, then any vertex v∈V(H) there six possible expressions, in terms parameters factor graphs, rooted product G∘vH. Additionally, characterize satisfy each these expressions.

Let $(X,d)$ be a metric space. A set $S\subseteq X$ is said to $k$-metric generator for $X$ if and only any pair of different points $u,v\in X$, there exist at least $k$ $w_1,w_2, \ldots w_k\in S$ such that $d(u,w_i)\ne d(v,w_i),\; \mbox{\rm all}\; i\in \{1, k\}.$ $\mathcal{R}_k(X)$ the generators $X$. The dimension $\dim_k(X)$ defined as $$\dim_k(X)=\inf\{|S|:\, S\in \mathcal{R}_k(X)\}.$$ Here, we discuss $(V,d_t)$, where $V$ vertices simple graph $G$ $d_t:V\times V\rightarrow...

In this paper, we show that the Italian domination number of every lexicographic product graph G ○ H can be expressed in terms five different parameters . These defined under following unified approach, which encompasses definition several well-known and introduces new ones. Let N ( v ) denote open neighbourhood ∈ V , let w = 0 1 …, l a vector nonnegative integers such ≥ We say function f : → {0, 1, } is -dominating if )) ∑ u i for vertex with The weight to ω -domination denoted by γ minimum...

In this paper we propose formulas for the distance between vertices of a generalized Sierpi?ski graph S(G, t) in terms base G. particular, deduce recursive formula an arbitrary vertex and extreme t), obtain two when is triangle-free. From these formulas, provide algorithms to compute t). addition, give explicit diameter radius tree.

Given a simple and connected graph $G=(V,E)$, positive integer $k$, set $S\subseteq V$ is said to be $k$-metric generator for $G$, if any pair of different vertices $u,v\in V$, there exist at least $k$ $w_1,w_2,\ldots,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$, every $i\in \{1,\ldots,k\}$, where $d_G(x,y)$ denotes the distance between $x$ $y$. The minimum cardinality dimension $G$. A $k$-adjacency $G$ two $x,y\in V(G)$ satisfy $|((N_G(x)\triangledown N_G(y))\cup\{x,y\})\cap S|\ge k$,...