Related papers: Equidistant dimension of Cartesian product graphs

An identifying code in a graph $G$ is a dominating set $C$ such that the closed neighborhood of each vertex in $G$ has a distinct intersection with $C$. In 2008, Gravier et al. determined the minimum cardinality of an identifying code of…

The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can…

A vertex $x$ in a graph $G$ resolves two vertices $u$, $v$ of $G$ if the distance between $u$ and $x$ is not equal to the distance between $v$ and $x$. A function $g$ from the vertex set of $G$ to $[0,1]$ is a resolving function of $G$ if…

A resolving set for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension $\mu(\Gamma)$ is the smallest size…

Let $G$ be a finite, simple, and undirected graph and let $S$ be a set of vertices of $G$. In the geodetic convexity, a set of vertices $S$ of a graph $G$ is convex if all vertices belonging to any shortest path between two vertices of $S$…

We will prove that some weighted graphs on the distance $k$-graph of hypercubes approximate the $q$-Hermite polynomial of a $q$-gaussian variable by providing an appropriate matrix model.

A resolving set for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension of $\Gamma$ is the smallest size of…

Given a graph $G$, a dominating set $D$ is a set of vertices such that any vertex in $G$ has at least one neighbor (or possibly itself) in $D$. A ${k}$-dominating multiset $D_k$ is a multiset of vertices such that any vertex in $G$ has at…

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it…

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner…

An axis-parallel $d$--dimensional box is a Cartesian product $R_1 \times R_2 \times ... \times R_d$ where $R_i$ (for $1 \le i \le d$) is a closed interval of the form $[a_i, b_i]$ on the real line. For a graph $G$, its \emph{boxicity}…

Let $G$ be a simple connected graph with vertex set $V(G)=\{v_{1}, v_{2}, \ldots, v_{n}\}$. The distance $d_G(v_i,v_j)$ between two vertices $v_i$ and $v_j$ of $G$ is the length of a shortest path between $v_i$ and $v_j$. The distance…

In this paper, we introduce a connection between two classical concepts of graph theory: \; metric dimension and distinguishing number. For a given graph $G$, let ${\rm dim}(G)$ and $D(G)$ represent its metric dimension and distinguishing…

An equidistant set in the Euclidean space consists of points having equal distances to both members of a given pair of sets, called focal sets. Having no effective formulas to compute the distance of a point and a set, it is hard to…

The relative Cayley graph of a group $G$ with respect to its proper subgroup $H$, is a graph whose vertices are elements of $G$ and two vertices $h\in H$ and $g\in G$ are adjacent if $g=hc$ for some $c\in C$, where $C$ is an inversed-closed…

The general position number ${\rm gp}(G)$ of a graph $G$ is the cardinality of a largest set of vertices $S$ such that no element of $S$ lies on a geodesic between two other elements of $S$. The complementary prism $G\overline{G}$ of $G$ is…

Equivalence relations on the edge set of a graph $G$ that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way…

A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$ than $v$. The boundary $\partial(G)$ of $G$ is the set of all of its boundary…

We calculate the metric dimension of the total graph of a direct product of finite commutative antinegative semirings with their sets of zero-divisors closed under addition.

The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $S\subseteq V(G)$ so that for any two vertices $x,y\in S$ there is a shortest $x,y$-path $P$ so that all internal vertices of $P$ are not in…