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Resolving sets are designed to locate an object in a network by measuring the distances to the object. However, if there are more than one object present in the network, this can lead to wrong conclusions. To overcome this problem, we introduce the concept of solid-resolving sets. In this paper, we study the structure and constructions of solid-resolving sets. In particular, we classify the forced vertices with respect to a solid-resolving set. We also give bounds on the solid-metric dimension utilizing concepts like the Dilworth number, the boundary of a graph, and locating-dominating sets. It is also shown that deciding whether there exists a solid-resolving set with a certain number of elements is an NP-complete problem.

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... We can also distinguish v 6 from X with another type of resolving sets introduced in [7]. The set S ⊆ V is a solid-resolving set of a graph G if for all v ∈ V and nonempty X ⊆ V we have D S (v) = D S (X). ...

... The set S ⊆ V (G) is an -solid-resolving set of G, if for all distinct nonempty sets X, Y ⊆ V (G) such that |X| ≤ we have D S (X) = D S (Y ). When = 1, the previous definition is exactly the same as the definition of a solid-resolving set in [7]. The set S 2 = {v 1 , v 2 , v 3 , v 4 , v 6 , v 8 , v 9 } is a 2-solidresolving set of H. ...

... On the Metric Dimensions for Sets of Vertices 5 Theorem 3 will be very useful throughout the article. This theorem also implies the corresponding result for = 1 in [7,Thm 2.2]. A somewhat similar result holds for { }-resolving sets as stated in the following lemma. ...

Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, {ℓ}-resolving sets were recently introduced. In this paper, we present new results regarding the {ℓ}-resolving sets of a graph. In addition to proving general results, we consider {2}-resolving sets in rook’s graphs and connect them to block designs. We also introduce the concept of ℓ-solid-resolving sets, which is a natural generalisation of solid-resolving sets. We prove some general bounds and characterisations for ℓ-solid-resolving sets and show how ℓ-solid- and {ℓ}-resolving sets are connected to each other. In the last part of the paper, we focus on the infinite graph family of flower snarks. We consider the ℓ-solid- and {ℓ}-metric dimensions of flower snarks. In two proofs regarding flower snarks, we use a new computer-aided reduction-like approach.

... We can also distinguish v6 from X with another type of resolving sets introduced in [7]. The set S ⊆ V is a solid-resolving set of a graph G if for all v ∈ V and nonempty X ⊆ V we have DS(v) = DS(X). ...

... When ℓ = 1, the previous definition is exactly the same as the definition of a solidresolving set in [7]. The set S2 = {v1, v2, v3, v4, v6, v8, v9} is a 2-solid-resolving set of H. ...

... Thus, the set S is an ℓ-solid-resolving set of G by Definition 2. Theorem 3 will be very useful throughout the article. This theorem also implies the corresponding result for ℓ = 1 in [7,Thm 2.2]. A somewhat similar result holds for {ℓ}resolving sets as stated in the following lemma. ...

Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, $\{\ell\}$-resolving sets were recently introduced. In this paper, we present new results regarding the $\{\ell\}$-resolving sets of a graph. In addition to proving general results, we consider $\{2\}$-resolving sets in rook's graphs and connect them to block designs. We also introduce the concept of $\ell$-solid-resolving sets, which is a natural generalisation of solid-resolving sets. We prove some general bounds and characterisations for $\ell$-solid-resolving sets and show how $\ell$-solid- and $\{\ell\}$-resolving sets are connected to each other. In the last part of the paper, we focus on the infinite graph family of flower snarks. We consider the $\ell$-solid- and $\{\ell\}$-metric dimensions of flower snarks. In two proofs regarding flower snarks, we use a new computer-aided reduction-like approach.

... In [7], such vertices were called basis forced vertices. Two variations of this idea, while considering the ℓ-solid resolving sets and the {ℓ}-resolving sets instead of the metric basis, were considered earlier in [5,6], respectively. ...

... Thus, the graph G does not contain configuration B with respect to the set S. C: By the previous case, there are no S-free threads at v 0 or v j . Furthermore, by (5), all the threads attached to some v i where i ∈ [1, j − 1] are of length at most g/2 − j − 1. ...

A metric basis in a graph $G$ is a smallest possible set $S$ of vertices of $G$, with the property that any two vertices of $G$ are uniquely recognized by using a vector of distances to the vertices in $S$. A strong metric basis is a variant of metric basis that represents a smallest possible set $S'$ of vertices of $G$ such that any two vertices $x,y$ of $G$ are uniquely recognized by a vertex $v\in S'$ by using either a shortest $x-v$ path that contains $y$, or a shortest $y-v$ path that contains $x$. Given a graph $G$, there exist sometimes some vertices of $G$ such that they forcedly belong to every metric basis or to every strong metric basis of $G$. Such vertices are called (resp. strong) basis forced vertices in $G$. It is natural to consider finding them, in order to find a (strong) metric basis in a graph. However, deciding about the existence of these vertices in arbitrary graphs is in general an NP-hard problem, which makes desirable the problem of searching for (strong) basis forced vertices in special graph classes. This article centers the attention in the class of unicyclic graphs. It is known that a unicyclic graph can have at most two basis forced vertices. In this sense, several results aimed to classify the unicyclic graphs according to the number of basis forced vertices they have are given in this work. On the other hand, with respect to the strong metric bases, it is proved in this work that unicyclic graphs can have as many strong basis forced vertices as we would require. Moreover, some characterizations of the unicyclic graphs concerning the existence or not of such vertices are given in the exposition as well.

... For some other variants of metric dimension, some related studies have continued. Examples of this are [9,10], where forced vertices were defined as vertices that are in every ℓ-solid-resolving set or {ℓ}-resolving set -not only the corresponding metric bases. In this sense, it is now our goal to retrieve such ideas, for the classical metric dimension, and present an exposition of combinatorial results aimed to describe structural properties of graphs containing vertices that either belong to every, or to no metric basis of it. ...

... In [9,10], forced vertices of ℓ-solid-resolving sets for all ℓ, and {ℓ}-resolving sets for ℓ ≥ 2 were characterised. These characterisations used the local properties of the vertices and their neighbourhoods. ...

A set R⊆V(G) is a resolving set of a graph G if for all distinct vertices v,u∈V(G) there exists an element r∈R such that d(r,v)≠d(r,u). The metric dimension dim(G) of the graph G is the cardinality of a smallest resolving set of G. A resolving set with cardinality dim(G) is called a metric basis of G. We consider vertices that are in all metric bases, and we call them basis forced vertices. We give several structural properties of sparse and dense graphs where basis forced vertices are present. In particular, we give bounds for the maximum number of edges in a graph containing basis forced vertices. Our bound is optimal whenever the number of basis forced vertices is even. Moreover, we provide a method of constructing fairly sparse graphs with basis forced vertices. We also study vertices which are in no metric basis in connection to cut-vertices and pendants. Furthermore, we show that deciding whether a vertex is in all metric bases is co-NP-hard, and deciding whether a vertex is in no metric basis is NP-hard.

... Resolving sets which identify other elements of the graphs: -edge resolving set [107] -a set such that any pair of edges of the graph is distinguished by the vertices of this set; -mixed resolving set [105] -a set such that any pair of elements (vertices or edges) of the graph is distinguished by the vertices of this set; -solid resolving set [82] -a set that uniquely identifies not only pairs of vertices but also pairs of subsets of vertices of the graph. ...

... A set S ⊂ V (G) is a solid-resolving set of G if for all vertices x ∈ V (G) and nonempty subsets Y ⊂ V (G), r(x|S) = r(Y |S) implies that Y = {x}. The minimum cardinality among all solidresolving sets of G is called the solid-metric dimension of G. Concepts above were introduced in [82] and further on generalized to {ℓ}-resolving sets and studied in [83,84]. ...

Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the highest interest for the research community dealing with several variants of metric dimension in graphs.

... For instance, the work of Lindström [17] is probably one of the oldest ones, and for some recent ones we suggest the works [7,18,24]. Surprisingly, for other related invariants there has been comparatively little research on hypercube graphs, although one can find some interesting recent results on this topic such as those that appeared in [5,7]. It is our goal to present some results on the close connections that exist among the metric, the edge metric and the mixed metric dimensions of hypercube graphs. ...

... In graph theory, the metric dimension problem (the problem statement and its background can be found in [1][2][3]) captivated numerous graph theorists because of its implications in various fields, which includes discovery of networks with security and verification [4], chemistry pharmaceutics in connection with the designing of drug [1], strategies of mastermind games [5], robot navigation [6], problems of coin weighing with its solutions [7], and connected joins in graphs [8]. Due to the empirical significance of this problem, for the last two decades, many researchers have tried to solve this problem by defining several versions of the metric dimension problem such as the fractional version [9], resolving domination [10], the version of doubly metric dimension [11], independent version [12], weighted version [13], the version of k−metric dimension [14], solid version [15], mixed version [16], edge version [17], local version [18], simultaneous version [19], connected version [20], and the strong version [8]. ...

The number of edges in a shortest walk (without repetition of vertices) from one vertex to another vertex of a connected graph G is known as the distance between them. For a vertex x and an edge e=ab in G, the minimum number from distances of x with a and b is said to be the distance between x and e. A vertex x is said to distinguish (resolves) two distinct edges e1 and e2 if the distance between x and e1 is different from the distance between x and e2. A set X of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex in X. The number of vertices in such a smallest set X is known as the edge metric dimension of G. In this article, we solve the edge metric dimension problem for certain classes of planar graphs.

... In [8] and [9], forced vertices of ℓ-solid-resolving sets for all ℓ, and {ℓ}-resolving sets for ℓ ≥ 2 were characterised. These characterisations used the local properties of the vertices and their neighbourhoods. ...

A set $R \subseteq V(G)$ is a resolving set of a graph $G$ if for all distinct vertices $v,u \in V(G)$ there exists an element $r \in R$ such that $d(r,v) \neq d(r,u)$. The metric dimension $\dim(G)$ of the graph $G$ is the minimum cardinality of a resolving set of $G$. A resolving set with cardinality $\dim(G)$ is called a metric basis of $G$. We consider vertices that are in all metric bases, and we call them basis forced vertices. We give several structural properties of sparse and dense graphs where basis forced vertices are present. In particular, we give bounds for the maximum number of edges in a graph containing basis forced vertices. Our bound is optimal whenever the number of basis forced vertices is even. Moreover, we provide a method of constructing fairly sparse graphs with basis forced vertices. We also study vertices which are in no metric basis in connection to cut-vertices and pendants. Furthermore, we show that deciding whether a vertex is in all metric bases is co-NP-hard, and deciding whether a vertex is in no metric basis is NP-hard.

... For instance, the work of Lindström [16] is probably one of the oldest ones, and for some recent ones we suggest the works [7,17,22]. Surprisingly, for other related invariants, there has been comparatively little research on hypercube graphs, although one can find some interesting recent results on this topic, such as those that appear in [5,7]. It is our goal to present some results on the closed connections that exist among the metric, edge metric and mixed metric dimensions of hypercube graphs. ...

The metric (resp. edge metric or mixed metric) dimension of a graph $G$, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of $G$ by using a vector of distances to this set. In this note we show two unexpected results on hypercube graphs. First, we show that the metric and edge metric dimension of $Q_d$ differ by only one for every integer $d$. In particular, if $d$ is odd, then the metric and edge metric dimensions of $Q_d$ are equal. Second, we prove that the metric and mixed metric dimensions of the hypercube $Q_d$ are equal for every $d \ge 3$. We conclude the paper by conjecturing that all these three types of metric dimensions of $Q_d$ are equal when $d$ is large enough.

... Next we consider boundary distance functions that are closely related to resolving sets of a graph, see [50,60,80] and their generalizations in [48,49]. ...

We study the Gel'fand's inverse boundary spectral problem of determining a finite weighted graph. Suppose that the set of vertices of the graph is a union of two disjoint sets: $X=B\cup G$, where $B$ is called the set of the boundary vertices and $G$ is called the set of the interior vertices. We consider the case where the vertices in the set $G$ and the edges connecting them are unknown. Assume that we are given the set $B$ and the pairs $(\lambda_j,\phi_j|_B)$, where $\lambda_j$ are the eigenvalues of the graph Laplacian and $\phi_j|_B$ are the values of the corresponding eigenfunctions at the vertices in $B$. We show that the graph structure, namely the unknown vertices in $G$ and the edges connecting them, along with the weights, can be uniquely determined from the given data, if every boundary vertex is connected to only one interior vertex and the graph satisfies the following property: any subset $S\subseteq G$ of cardinality $|S|\geqslant 2$ contains two extreme points. A point $x\in S$ is called an extreme point of $S$ if there exists a point $z\in B$ such that $x$ is the unique nearest point in $S$ from $z$ with respect to the graph distance. This property is valid for several standard types of lattices and their perturbations.

... Because of these practical significances of this problem, from the last two decades, numerous researchers identified vertices of a graph by considering the metricrelated well-known concept of the metric dimension [2,7,8]. Later on, many researchers extended the study of this concept by defining its several variations including the fractional metric dimension [9], the resolving domination [10], the doubly metric dimension [11], the independent metric dimension [12], the weighted metric dimension [13], the k-metric dimension [14], the solid metric dimension [15], the mixed metric dimension [16], the local metric dimension [17], the simultaneous metric dimension [18], the strong metric dimension [5], and the connected metric dimension [19]. ...

A shortest path between two vertices u and v in a connected graph G is a u−v geodesic. A vertex w of G performs the geodesic identification for the vertices in a pair u,v if either v belongs to a u−w geodesic or u belongs to a v−w geodesic. The minimum number of vertices performing the geodesic identification for each pair of vertices in G is called the strong metric dimension of G. In this paper, we solve the strong metric dimension problem for three convex plane graphs by performing the geodesic identification of their vertices.

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

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded.

Due to its fast, dynamic, and distributed growth process, it is hard to obtain an accurate map of the Internet. In many cases, such a map-representing the structure of the Internet as a graph with nodes and links-is a prerequisite when investigating properties of the Internet. A common way to obtain such maps is to make certain local measurements at a small subset of the nodes, and then to combine these in order to "discover" (an approximation of) the actual graph. Each of these measurements is potentially quite costly. It is thus a natural objective to minimize the number of measurements which still discover the whole graph. We formalize this problem as a combinatorial optimization problem and consider it for two different models characterized by different types of measurements. We give several upper and lower bounds on the competitive ratio (for the online network discovery problem) and the approximation ratio (for the offline network verification problem) in both models. Furthermore, for one of the two models, we compare four simple greedy strategies in an experimental analysis

Identifying codes and locating-dominating codes have been designed for locating irregularities in sensor networks. In both cases, we can locate only one irregularity and cannot even detect multiple ones. To overcome this issue, self-identifying codes have been introduced which can locate one irregularity and detect multiple ones. In this paper, we define two new classes of locating-dominating codes which have similar properties. These new locating-dominating codes as well as the regular ones are then more closely studied in the rook’s graphs and binary Hamming spaces.
In the rook’s graphs, we present optimal codes, i.e., codes with the smallest possible cardinalities, for regular location-domination as well as for the two new classes. In the binary Hamming spaces, we present lower bounds and constructions for the new classes of codes; in some cases, the constructions are optimal. Moreover, one of the obtained lower bounds improves the bound of Honkala et al. (2004) on codes for locating multiple irregularities.
Besides studying the new classes of codes, we also present record-breaking constructions for regular locating-dominating codes. In particular, we present a locating-dominating code in the binary Hamming space of length 11 with 320 vertices improving the earlier bound of 352; the best known lower bound for such code is 309 by Honkala et al. (2004).

A set of vertices S is a resolving set in a graph if each vertex has a unique array of distances to the vertices of S. The natural problem of finding the smallest cardinality of a resolving set in a graph has been widely studied over the years. In this paper, we wish to resolve a set of vertices (up to ℓ vertices) instead of just one vertex with the aid of the array of distances. The smallest cardinality of a set S resolving at most ℓ vertices is called ℓ-set-metric dimension. We study the problem of the ℓ-set-metric dimension in two infinite classes of graphs, namely, the two dimensional grid graphs and the n-dimensional binary hypercubes.

Connections between several parameters of a graph and the structure of an associated preorder are examined. The Dilworth number of the preorder appears to have a particular importance.

A set S of vertices of a graph G is said to be a k -metric generator for G if for any u,v∈V(G)u,v∈V(G), u≠vu≠v, there exists Suv⊆SSuv⊆S such that |Suv|≥k|Suv|≥k and for every w∈Suvw∈Suv, dG(u,w)≠dG(v,w)dG(u,w)≠dG(v,w). A metric generator of minimum cardinality is called a k-metric basis and its cardinality the k-metric dimension of G. We give a necessary and sufficient condition for the existence of a k-metric basis of a graph and we obtain several results on the k-metric dimension.

Locating-dominating sets are of interest in safeguard applications of graphical models of facilities. A subset S of the vertex set V of a graph G is a dominating set if each vertex u ϵ V - S is adjacent to at least one vertex in S. For each v in V - S let S(v) denote the set of vertices in S which are adjacent to v. A dominating set S is defined to be “locating” if for any two vertices v and w in V - S one has S(v) ≠ S(w). Sharp bounds on the cardinality of locating-dominating sets for arbitrary graphs on p vertices and for trees on p vertices are given, and a linear (that is O(P)) algorithm for finding a minimum cardinality locating-dominating set in an acyclic graph is presented.

Navigation can be studied in a graph-structured framework in which the navigating agent (which we shall assume to be a point robot) moves from node to node of a “graph space”. The robot can locate itself by the presence of distinctively labeled “landmark” nodes in the graph space. For a robot navigating in Euclidean space, visual detection of a distinctive landmark provides information about the direction to the landmark, and allows the robot to determine its position by triangulation. On a graph, however, there is neither the concept of direction nor that of visibility. Instead, we shall assume that a robot navigating on a graph can sense the distances to a set of landmarks.Evidently, if the robot knows its distances to a sufficiently large set of landmarks, its position on the graph is uniquely determined. This suggests the following problem: given a graph, what are the fewest number of landmarks needed, and where should they be located, so that the distances to the landmarks uniquely determine the robot's position on the graph? This is actually a classical problem about metric spaces. A minimum set of landmarks which uniquely determine the robot's position is called a “metric basis”, and the minimum number of landmarks is called the “metric dimension” of the graph. In this paper we present some results about this problem. Our main new results are that the metric dimension of a graph with n nodes can be approximated in polynomial time within a factor of O(logn), and some properties of graphs with metric dimension two.

Assume that G=(V,E) is an undirected graph, and C⊆V. For every v∈V, we denote , where d(u,v) denotes the number of edges on any shortest path from u to v. For every F⊆V, we denote Ir(F)=⋃v∈FIr(v). We study codes C with the property that if Ir(F)=Ir(F′) and F≠F′, then both F and F′ have size at least l+1. Such codes can be used in the maintenance of multiprocessor architectures. We consider the cases when G is the infinite square or king grid, infinite triangular lattice or hexagonal mesh, or a binary hypercube.

For an ordered subset W={w1,w2,…,wk} of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector , d(v,w2),…,d(v,wk)). The set W is a resolving set for G if r(u|W)=r(v|W) implies that u=v for all pairs u,v of vertices of G. The metric dimension dim(G) of G is the minimum cardinality of a resolving set for G. Bounds on dim(G) are presented in terms of the order and the diameter of G. All connected graphs of order n having dimension 1,n−2, or n−1 are determined. A new proof for the dimension of a tree is also presented. From this result sharp bounds on the metric dimension of unicyclic graphs are established. It is shown that dim(H)⩽dim(H×K2)⩽dim(H)+1 for every connected graph H. Moreover, it is shown that for every positive real number ε, there exists a connected graph G and a connected induced subgraph H of G such that dim(G)/dim(H)<ε.

We study generators of metric spaces—sets of points with the property that every point of the space is uniquely determined by the distances from their elements. Such generators put a light on seemingly different kinds of problems in combinatorics that are not directly related to metric spaces. The two applications we present concern combinatorial search: problems on false coins known from the borderline of extremal combinatorics and information theory; and a problem known from combinatorial optimization—connected joins in graphs.
We use results on the detection of false coins to approximate the metric dimension (minimum size of a generator for the metric space defined by the distances) of some particular graphs for which the problem was known and open. In the opposite direction, using metric generators, we show that the existence of connected joins in graphs can be solved in polynomial time, a problem asked in a survey paper of Frank. On the negative side we prove that the minimization of the number of components of a join is NP-hard.
We further explore the metric dimension with some problems. The main problem we are led to is how to extend an isometry given on a metric generator of a metric space.

The distance d(u,v) between two vertices u and v in a nontrivial connected graph G is the length of a shortest u–v path in G. For a vertex v of G, the eccentricity e(v) is the distance between v and a vertex farthest from v. A vertex v of G is a peripheral vertex if e(v) is the diameter of G. The subgraph of G induced by its peripheral vertices is the periphery Per(G) of G. A vertex u of G is an eccentric vertex of a vertex v if d(u,v)=e(v). A vertex x is an eccentric vertex of G if x is an eccentric vertex of some vertex of G. The subgraph of G induced by its eccentric vertices is the eccentric subgraph Ecc(G) of G. A vertex u of G is a boundary vertex of a vertex v if d(w,v)⩽d(u,v) for all w∈N(u). A vertex u is a boundary vertex of G if u is a boundary vertex of some vertex of G. The subgraph of G induced by its boundary vertices is the boundary ∂(G) of G. A graph H is a boundary graph if H=∂(G) for some graph G. We study the relationship among the periphery, eccentric subgraph, and boundary of a connected graph and establish a characterization of all boundary graphs. It is shown that for each triple a,b,c of integers with 2⩽a⩽b⩽c, there is a connected graph G such that Per(G) has order a, Ecc(G) has order b, and ∂(G) has order c. Moreover, for each triple r,s,t of rational numbers with 0

Let u, v ∈ V be two vertices of a connected graph G.T he vertexv is said to be a boundary vertex of u if no neighbor of v is further away from u than v .T he boundary of a graph is the set of all its boundary vertices. In this work, we present a number of properties of the boundary of a graph under different points of view: (1) a realization theorem involving different types of boundary vertex sets: extreme set, periphery, contour, and the whole boundary; (2) the boundary is an edge-geodetic set, and the contour is a monophonic set; (3) the boundary is a resolving set.

A decomposition theorem for partially ordered sets

- Dilworth