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# The solid-metric dimension

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## Abstract

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. ...
Article
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. ...
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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. ...
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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. ...
Article
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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]. ...
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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. ...
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... 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]. ...
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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. ...
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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. ...
Preprint
Full-text available
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]. ...
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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]. ...
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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.
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A decomposition theorem for partially ordered sets
• Dilworth