l-Clique Metric Dimension of Graphs

Abstract

For an ordered non-empty subset $S=\{v_1,\ldots , v_k\}$ of vertices in a connected graph G and an l-clique $V'$ of G, the l-clique metric S-representation of $V'$ is the vector $r^l_G(V'|S) = (d_G(V',v_1), \ldots , d_G(V',v_k))\,$ where $d_G(V',v_i)=\min \{d_G(v,v_i): v\in V'\}$. A non-empty subset S of V(G) is an l-clique metric generator for G if all l-cliques of G have pairwise different l-clique metric S-representations. An l-clique metric generator of smallest order is an l-clique metric basis for G, its order being the l-clique metric dimension (l-CMD for short) $\mathrm{cdim}_l(G)$ of G. In this paper, we propose this concept as an extension of the 1-clique metric dimension which is known as the metric dimension, and also study some its properties. Moreover, l-CMD for $\Gamma ({\mathbb {Z}}_n)$ and the corona product of two graphs is investigated. Furthermore, we prove that computing the l-CMD of connected graphs is NP-hard and present an integer linear programming model for finding this parameter.

1 Introduction

All graphs considered in this paper are undirected and simple.

If $u, v\in V(G)$, then $d_G(u,v)$ denotes the number of edges on a shortest u, v-path in G. A clique $V'$ is a subset of vertices of a graph such that every two distinct vertices in the clique are adjacent. Also, $V'$ is called an l-clique if $|V'|=l$. For a vertex u and an l-clique $V'$ of G, the distance between $V'$ and u, denoted by $d_G(u,V')$, is defined as $\min \{d_G(u,v): v\in V'\}$; in other words, $d_G(u,V')=\min \{d_G(u,v): v\in V'\}$.

For an ordered non-empty subset $S=\{v_1,\ldots , v_k\}$ of vertices in a connected graph G and an l-clique $V'$ of G, the l -clique metric S-representation of $V'$ is the vector $r^l_G(V'|S) = (d_G(V',v_1), \ldots , d_G(V',v_k))$. A non-empty subset S of V(G) is an l-clique metric generator for G if all l-cliques of G have pairwise different l-clique metric S-representations. l-Clique metric generators for special cases $l=1$ and $l=2$ are known as metric generator and edge metric generator, respectively. An l-clique metric generator of smallest order is an l-clique metric basis for G, its order being the l-clique metric dimension (l-CMD for short) $\mathrm{cdim}_l(G)$ of G.

Recall that the special case 1-clique metric dimension is called the metric dimension and denoted by $\mathrm{dim}(G)$ and also the special case 2-clique metric dimension is called the edge metric dimension and denoted by $\mathrm{dim}_e(G)$.

The concept of metric dimension was first introduced by Slater [21]. Since then lots of work has been done on this topic because of its wide range of applications in modeling of real world problems [13, 15]. For instance, Garey and Johnson [11], and Epstein et al. [10] studied NP-hardness of computing of metric dimension. Also, this invariant was investigated over the Cartesian product of graphs in [5], over the lexicographic product of graphs in [19], over the deleted lexicographic product of graphs in [9], and over the hierarchical product of graphs in [23]. Kelenc et al. [14] introduced the concept of edge metric dimension. In the present work, we expand the concept of metric dimension as l-clique metric dimension where l is a natural number. Note that in [12] resolving sets locate up to some fixed l, $l\ge 1$, vertices in a graph, while here resolving sets locate the l-cliques of a graph. The first section of this paper is dedicated to some properties of this parameter of graphs. In the second section, we compute l-CMD for $\Gamma ({\mathbb {Z}}_n)$. We also obtain the exact value of l-CMD of corona product of two graphs in the third section. [11, 14] showed the NP-completeness of l-CMD problems for $l=1$ and $l=2$, respectively. We prove the NP-completeness of l-CMD problems for $l\ge 3$ in the last section.

Throughout this paper, our notation is standard and taken mainly from [2].

2 Basic Results

In this section, we present some basic results on the l-clique metric dimension.

The following proposition gives the l-CMD of the complete graph $K_n$.

Proposition 2.1

Let $n \geqslant 2$. We have

$$\begin{aligned} \mathrm{cdim}_l(K_n) = \left\{ \begin{array}{ll} 1 &{}\quad l=n\\ n-1 &{}\quad \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

Proof

If $l=n$, then clearly $\mathrm{cdim}_l(K_n)=1$. Let $l \in \{1,2\}$. Then, by [14,  Remark 1], we have $\mathrm {edim}(K_n)=\mathrm {dim}(K_n)=n-1$. Hence, in this situation, $\mathrm{cdim}_l(K_n)=\mathrm {edim}(K_n)=\mathrm {dim}(K_n)=n-1$. So we assume that $3 \leqslant l \leqslant n-1$ and $n \geqslant 4$. Let S be a minimal l-clique metric generator of $K_n$. If $|S|\leqslant n-2$, then there exist two distinct vertices $x,y \in K_n {\setminus } S$. Consider two l-cliques $L_1$ and $L_2$ such that $x\in L_1$, $y \in L_2$ and $L_1 {\setminus } \{x\}=L_2{\setminus } \{y\}$. Then one can see that the l-clique metric S-representations of $L_1$ and $L_2$ are the same, which is impossible. Now, let $S\subseteq V(K_n)$ with $|S|=n-1$. Then, in this situation, for every two distinct cliques $L_1$ and $L_2$, there exists $s \in S$ such that $s \in L_1 {\setminus } L_2$. Therefore the component which is corresponding to s in the l-clique metric S-representations of $L_1$ and $L_2$ is 0 and 1, respectively, which implies that S is an l-clique metric generator for $K_n$. Hence $\mathrm{cdim}_l(K_n)=n-1$. $\square$

Recall that the wheel graph $W_{1,n}$ is the graph obtained from a cycle $C_n$ and the graph $K_1$ by adding all the edges between the vertex of $K_1$ and every vertex of $C_n$.

The least integer greater than or equal to a number m is denoted by $\lceil m \rceil$. Also, greatest integer less than or equal to a number m is denoted by $\lfloor m \rfloor$.

In the following proposition, we investigate the l-CMD $\mathrm{cdim}_l(W_{1,n})$. Note that if $l=1$, then $\mathrm{cdim}_l(W_{1,n})=\mathrm {dim}(W_{1,n})$, which is determined in [3], as follows.

$$\begin{aligned} \mathrm {dim}(W_{1,n}) = \left\{ \begin{array}{ll} 3 &{}\quad n=3,6\\ 2 &{}\quad n=4,5\\ \lfloor \frac{2n+2}{5}\rfloor &{}\quad n\geqslant 6. \end{array} \right. \end{aligned}$$

Also, if $l=2$, then $\mathrm{cdim}_l(W_{1,n})=\mathrm {edim}(W_{1,n})$, which is

$$\begin{aligned} \mathrm {edim}(W_{1,n}) = \left\{ \begin{array}{ll} n &{}\quad n=3,4\\ n-1 &{}\quad n\geqslant 5, \end{array} \right. \end{aligned}$$

see [14].

Proposition 2.2

Let $W_{1,n}$ be a wheel graph. Then

$$\begin{aligned} \mathrm{cdim}_3(W_{1,n}) = \left\{ \begin{array}{ll} 3 &{}\quad n=3\\ n-\lceil \frac{n}{3}\rceil &{}\quad n\geqslant 4. \end{array} \right. \end{aligned}$$

Proof

By Proposition 2.1, we have $\mathrm{cdim}_3(W_{1,3}) =\mathrm{cdim}_3(K_4)=3$ and $\mathrm{cdim}_4(W_{1,3}) =\mathrm{cdim}_4(K_4)=1$. So assume that $n\geqslant 4$. Let $\{g_1,g_2, \dots , g_n\}$ be the vertices of degree 3 in $W_{1,n}$. Clearly for each two distinct triangles $L_1$ and $L_2$ in $W_{1,n}$, either there exists $1 \leqslant i \leqslant n$ such that $L_1$ and $L_2$ have the common vertex $g_i$, or $L_1$ and $L_2$ have no common vertices from the set $\{g_1,g_2, \dots , g_n\}$. In both of the situations, one can easily see that $L_1$ and $L_2$ have the same 3-clique metric S-representations if and only if their non-common vertices do not belong to S, where $S \subseteq V(W_{1,n})$. Now let S be a 3-clique metric basis of $W_{1,n}$. Clearly $S \subseteq \{g_1,g_2, \dots , g_n\}$. We consider the following cases.

Case 1 $n=3k$, where $k \geqslant 2$. Let S be a 3-clique metric basis of $W_{1,n}$. If there are two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$, say $g_2$ and $g_3$, such that $g_2,g_3 \notin S$, then we should have $g_4,g_5\in S$ and $g_{3\lfloor \frac{n}{3}\rfloor }, g_1 \in S$. So, if $k=2$, then $\mathrm{cdim}_3(W_{1,6}) =4$. Let $k>2$. Since S is a 3-clique metric basis, without loss of generality, we may assume that $g_6 \notin S , g_7,g_8\in S, \dots ,g_{3\lfloor \frac{n}{3}\rfloor -3}\notin S, g_{3\lfloor \frac{n}{3}\rfloor -2}, g_{3\lfloor \frac{n}{3}\rfloor -1} \in S$. Therefore, in this situation, $|S|=n-\lceil \frac{n}{3}\rceil$.

Now, assume that there exists a 3-clique metric basis of $W_{1,n}$, say S, such that for any two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$ at least one of them belongs to S. Without loss of generality, assume that $g_3 \notin S$. Since S is a 3-clique metric basis, we may assume that

$$\begin{aligned} S=\{g_1,g_2, \dots , g_n\} {\setminus } \{g_3,g_6,\dots ,g_{3i}, \dots ,g_{3\lfloor \frac{n}{3}\rfloor }\}, \end{aligned}$$

where $1 \leqslant i \leqslant \lfloor \frac{n}{3}\rfloor$. Clearly, in this situation we again have $|S|=n-\lceil \frac{n}{3}\rceil$.

Note that in either of the above situationes, by the structure that we obtain for a 3-clique metric basis of $W_{1,n}$, it is easy to see that any subset of $\{g_1,g_2, \dots , g_n\}$ with less that $n-\lceil \frac{n}{3}\rceil$ elements is not a 3-clique metric generator of $W_{1,n}$. Therefore, in this case the 3-CMD of $W_{1,n}$ is equal to $n-\lceil \frac{n}{3}\rceil$.

Case 2 $n=3k+1$ or $n=3k+2$, where $k \geqslant 1$. First we show that for any 3-clique metric basis of $W_{1,n}$, say S, there exist two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$ such that they do not belong to S. Assume on the contrary that for any two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$, at least one of them belongs to S. Without loss of generality, we may assume that $g_3 \notin S$. Since S is a 3-clique metric basis, we may assume that

$$\begin{aligned} S=\{g_1,g_2, \dots , g_n\} {\setminus } \{g_3,g_6,\dots ,g_{3i},\dots ,g_{3\lfloor \frac{n}{3}\rfloor }\}, \end{aligned}$$

where $1 \leqslant i \leqslant \lfloor \frac{n}{3}\rfloor$. Now consider the set $S'=S {\setminus } \{g_2\}$. One can easily see that $S'$ is a 3-clique metric generator of $W_{1,n}$ with $|S'|<|S|$, which is a contradiction.

Now let S be a 3-clique metric basis of $W_{1,n}$. Then there are two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$, say $g_2$ and $g_3$, such that $g_2,g_3 \notin S$. By using a similar discussion as we used in Case 1, we obtain that

$$\begin{aligned} S=\{g_1,g_2, \dots , g_n\} {\setminus } \{g_2,g_3,g_6,\dots ,g_{3i},\dots ,g_{3\lfloor \frac{n}{3}\rfloor }\} \end{aligned}$$

where $1 \leqslant i \leqslant \lfloor \frac{n}{3}\rfloor$ and $|S|=n-\lceil \frac{n}{3}\rceil$. Also, by the structure that we obtain for S, it is easy to see that any subset of $\{g_1,g_2, \dots , g_n\}$ with less that $n-\lceil \frac{n}{3}\rceil$ elements, is not a 3-clique metric generator of $W_{1,n}$.

Therefore we have $\mathrm{cdim}_3(W_{1,n}) =n-\lceil \frac{n}{3}\rceil$, when $n \geqslant 4$. $\square$

Similarly to the wheel graph, the fan graph, which is denoted by $F_{1,n}$, is the graph that is obtained from a path $P_n$ and the graph $K_1$ by adding all the edges between the vertex of $K_1$ and every vertex of $P_n$. In [4, 14], $\mathrm {dim}(F_{1,n})$ and $\mathrm {edim}(F_{1,n})$ are determined as follows:

$$\begin{aligned} \mathrm {dim}(F_{1,n}) = \left\{ \begin{array}{ll} 1 &{}\quad n=1\\ 2 &{}\quad n=2,3\\ 3 &{}\quad n=6\\ \lfloor \frac{2n+2}{5}\rfloor &{}\quad \mathrm{otherwise} \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \mathrm {edim}(F_{1,n}) = \left\{ \begin{array}{ll} n &{}\quad n=1,2,3\\ n-1 &{}\quad n\geqslant 4. \end{array} \right. \end{aligned}$$

In the following proposition, we investigate the l-CMD of $F_{1,n}$ in the case that $l=3$.

Proposition 2.3

For the fan graph $F_{1,n}$ we have

$$\begin{aligned} \mathrm{cdim}_3(F_{1,n}) = \left\{ \begin{array}{ll} 1 &{}\quad n=1,2\\ n-\lceil \frac{n}{3}\rceil -1 &{}\quad n=3k, 3k+2 \; \mathrm {for} \; k\geqslant 1 \\ n-\lceil \frac{n}{3}\rceil &{}\quad \mathrm{otherwise}. \end{array} \right. \end{aligned}$$

Proof

Clearly if $n\in \{1,2,3\}$, we have $\mathrm{cdim}_3(F_{1,n}) =1$. Let $\{g_1, g_2, \dots , g_n\}$ be the vertices of the path $P_n$ in the structure of $F_{1,n}$. Note that for each two distinct triangles $L_1$ and $L_2$ in $F_{1,n}$, they have the same 3-clique metric S-representations if and only if their non-common vertices do not belong to S, where $S\subseteq V(F_{1,n})$. Also clearly each 3-clique metric basis of $F_{1,n}$ is a subset of $\{g_1, g_2, \dots , g_n\}$. Now we have the following cases:

Case 1 $n=3k$, where $k \geqslant 2$. First we show that for any 3-clique metric basis of $F_{1,n}$, say S, there exist two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$ such that they do not belong to S. Assume on the contrary that for any two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$, at least one of them belongs to S. If $g_1 \notin S$, then by using a similar method as we used in the proof of Proposition 2.2, we get that

$$\begin{aligned} S=\{g_1,g_2, \dots , g_n\} {\setminus } \{g_1,g_4,\dots ,g_{3i+1},\dots ,g_{3\lfloor \frac{n}{3}\rfloor -2}\}, \end{aligned}$$

where $0 \leqslant i \leqslant \lfloor \frac{n}{3}\rfloor -1$. But one can easily see that the set $S'=S {\setminus } \{g_{3\lfloor \frac{n}{3}\rfloor -1}\}$ is a 3-clique metric generator of $F_{1,n}$ with $|S'|<|S|$, which is a contradiction. Now, let $g_1 \in S$. Then we may assume that

$$\begin{aligned} S=\{g_1,g_2, \dots , g_n\} {\setminus } \{g_2,g_5,\dots ,g_{3i+2},\dots ,g_{3\lfloor \frac{n}{3}\rfloor -1}\}, \end{aligned}$$

where $0 \leqslant i \leqslant \lfloor \frac{n}{3}\rfloor -1$. Again we see that the set $S'=S {\setminus } \{g_{3\lfloor \frac{n}{3}\rfloor }\}$ is a 3-clique metric generator of $F_{1,n}$ with $|S'|<|S|$, which is a contradiction. Therefore for any 3-clique metric basis of $F_{1,n}$, say S, there exist two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$ such that they do not belong to S. Now it is easy to see that

$$\begin{aligned} S=\{g_1,g_2, \dots , g_n\} {\setminus } \{g_1,g_2,g_5,\dots ,g_{3i+2},\dots ,g_{3\lfloor \frac{n}{3}\rfloor -1}\}, \end{aligned}$$

where $0 \leqslant i \leqslant \lfloor \frac{n}{3}\rfloor -1$ is a 3-clique metric generator of $F_{1,n}$, and any subset of $\{g_1,g_2, \dots , g_n\}$ with cardinality less than $|S|=n-\lceil \frac{n}{3}\rceil -1$ is not a 3-clique metric generator for $F_{1,n}$. Hence in this case we have $\mathrm{cdim}_3(F_{1,n}) =n-\lceil \frac{n}{3}\rceil -1$.

Case 2 $n=3k+1$, where $k \geqslant 1$. Let S be a 3-clique metric basis. First assume that for any two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$, at least one of them belongs to S. If $g_1 \in S$, then if $g_2 \notin S$, then $S {\setminus } \{g_1\}$ is a 3-clique metric generator with less than |S| elements which is impossible. Also if $g_2 \in S$, then $S {\setminus } \{g_2\}$ is a 3-clique metric generator with less than |S| elements which is again impossible. So we have $g_1 \notin S$. In this situation, one can easily see that

$$\begin{aligned} S=\{g_1,g_2, \dots , g_n\} {\setminus } \{g_1,g_4,\dots ,g_{3i+1}, \dots ,g_{3\lfloor \frac{n}{3}\rfloor +1}\}, \end{aligned}$$

where $0 \leqslant i \leqslant \lfloor \frac{n}{3}\rfloor$ is a 3-clique metric basis for $F_{1,n}$, with $|S|=n-\lceil \frac{n}{3}\rceil$. Now, suppose that there exist two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$ such that they do not belong to S. In this situation, we again have $|S|=n-\lceil \frac{n}{3}\rceil$. Therefore in this case we have $\mathrm{cdim}_3(F_{1,n}) =n-\lceil \frac{n}{3}\rceil$.

Case 3 $n=3k+2$, where $k \geqslant 1$. Similar to Case 1, we can see that for any 3-clique metric basis of $F_{1,n}$, say S, there exist two adjacent vertices of the set $\{g_1,g_2, \dots , g_n\}$ such that they do not belong to S. Now one can easily see that

$$\begin{aligned} S=\{g_1,g_2, \dots , g_n\} {\setminus } \{g_1,g_2,g_5,\dots ,g_{3i+2},\dots ,g_{3\lfloor \frac{n}{3}\rfloor +2}\}, \end{aligned}$$

where $0 \leqslant i \leqslant \lfloor \frac{n}{3}\rfloor$, is a 3-clique metric generator of $F_{1,n}$, and any subset of $\{g_1,g_2, \dots , g_n\}$ with cardinality less than $|S|=n-\lceil \frac{n}{3}\rceil -1$ is not a 3-clique metric generator for $F_{1,n}$. Hence in this case we have $\mathrm{cdim}_3(F_{1,n}) =n-\lceil \frac{n}{3}\rceil -1$. $\square$

Proposition 2.4

Let G be a graph with n vertices such that the number of its l-cliques are t. Then if $t \geqslant 2$, we have $\mathrm{cdim}_l(G) \leqslant \mathrm {min}\{n, \begin{pmatrix} t \\ 2 \end{pmatrix}\}.$ Otherwise $\mathrm{cdim}_l(G) =1$.

Proof

If $l=1$ or $t \leqslant 1$, then clearly we are done. So assume that $l \geqslant 2$. Let $L_1,L_2, \dots ,L_t$ be the l-cliques of G. For each $1 \leqslant i < j \leqslant n$, consider a vertex $x_{i,j}$ which belongs to $L_i {\setminus } L_j$. Let $S=\{x_{i,j} \mid 1 \leqslant i < j \leqslant n\}$. Now one can see that S is an l-clique metric generator for G and $|S| \leqslant \begin{pmatrix} t \\ 2 \end{pmatrix}$. Hence the result holds. $\square$

The next corollary follows from Proposition 2.4.

Corollary 2.5

Let G be a graph with at most two l-cliques. Then $\mathrm{cdim}_l(G) =1$.

Proposition 2.6

Let G be a graph with n vertices and $L_1,L_2, \dots ,L_t$ be the l-cliques of G such that $L_i \nsubseteq \bigcup _{i \ne j, j=1}^t L_j$, for $1 \leqslant i \leqslant t-1$. Then $\mathrm{cdim}_l(G) \leqslant t-1$.

Proof

Let $x_i \in L_i {\setminus } \bigcup _{i \ne j, j=1}^t L_j$, for $1 \leqslant i \leqslant t-1$. Set $S=\{x_i \mid 1 \leqslant i \leqslant t-1\}$. Then the ith component of the l-clique metric S-representation of $L_j$ is zero if and only if $i=j$, for $1 \leqslant i \leqslant t-1$. Moreover, none of the components of the l-clique metric S-representation of $L_t$ is zero. Hence S is an l-clique metric generator of G, and so $\mathrm{cdim}_l(G) \leqslant t-1$. $\square$

If we consider disconnected graphs, then l-CMD could be easily defined by considering the distance between two vertices in two different components as infinite. In fact we have the following result.

Remark 2.7

Let G be a disconnected graph with components $G_1, \dots , G_r$. If $I=\{i \mid G_i \, \mathrm {has} \, \mathrm {one} \, l\mathrm {-clique}\}$ and $J=\{i \mid G_i \, \mathrm {has} \, \mathrm {at} \, \mathrm {least} \, \mathrm {two} \, l\mathrm {-cliques}\}$, then

$$\begin{aligned} \mathrm{cdim}_l(G) =\sum _{i\in J} \mathrm{cdim}_l(G_i)+\left\{ \begin{array}{ll} 0 &{}\quad |I| \leqslant 1 \\ |I|-1 &{}\quad |I|>1 \end{array}. \right. \end{aligned}$$

Recall that for two graphs $H_1$ and $H_2$ with disjoint vertex sets, the join $H_1 \vee H_2$ of the graphs $H_1$ and $H_2$ is the graph obtained from the union of $H_1$ and $H_2$ by adding new edges from each vertex of $H_1$ to every vertex of $H_2$. The concept of join graph is generalized (in [17], it is called as a generalized composition graph). Assume that G is a graph on k vertices with $V(G)=\{v_1,v_2,\dots ,v_k\}$, and let $H_1,H_2,\dots ,H_k$ be k pairwise disjoint graphs. The G-generalized join graph $G[H_1,H_2,\dots ,H_k]$ of $H_1,H_2, \dots ,H_k$ is the graph formed by replacing each vertex $v_i$ of G by the graph $H_i$ and then joining each vertex of $H_i$ to each vertex of $H_j$ whenever $v_i\sim v_j$ in the graph G. Now, if the graph G consists of two adjacent vertices, then the G-generalized join graph $G[H_1,H_2]$ coincides with the join $H_1 \vee H_2$ of the graphs $H_1$ and $H_2$.

Note that in the rest of this section, we assume that there exists at least a nontrivial $H_i$, with $1 \leqslant i \leqslant k$, in $G[H_1,H_2,\dots ,H_k]$.

In the following proposition, we study the l-CMD of the G-generalized join graph $G[H_1,H_2,\dots ,H_k]$, in the case that $H_i$’s are empty graphs.

Proposition 2.8

Assume that G is a connected graph on k vertices with $V(G)=\{v_1,v_2,\dots ,v_k\}$, and let $H_1,H_2,\dots ,H_k$ be k pairwise disjoint empty graphs. If $\{v_1,v_2,\dots ,v_t\}$, where $0 \leqslant t \leqslant k$ are the vertices in G such that each of them belongs to an l-clique, then

$$\begin{aligned} \sum _{i=1}^t|V(H_i)|-t \leqslant \mathrm{cdim}_l(G[H_1,H_2,\dots ,H_k]) \leqslant \mathrm{cdim}_l(G)+\sum _{i=1}^t|V(H_i)|-t. \end{aligned}$$

Proof

Let $\{v_{i_1},v_{i_2},\dots ,v_{i_t}\}$, where $0 \leqslant t \leqslant k$ be the vertices in G such that each of them belongs to at least one l-clique. If $t=0$, then $\mathrm{cdim}_l(G[H_1,H_2,\dots ,H_k]) =\mathrm{cdim}_l(G)=1$. So assume that $t>0$. Let $h_1, \dots , h_t$ be arbitrary vertices in $H_1,\dots ,H_t$, respectively. Assume that S is an l-clique metric generator of the graph $G[H_1,H_2,\dots ,H_k]$. For each $1 \leqslant i \leqslant t$, we show that $V(H_i) {\setminus } \{h_i\} \subseteq S$. Suppose on the contrary that there exists $h'_i \in V(H_i)$ with $h'_i\ne h_i$ such that $h'_i \notin S$. Now consider two l-cliques $L_1$ and $L_2$ such that $h_i$ is a vertex of $L_1$, $h'_i$ is a vertex of $L_2$ and $L_1 {\setminus } \{h_i\}=L_2 {\setminus } \{h'_i\}$. Now, one can see that the l-clique metric S-representations of $L_1$ and $L_2$ are the same, which is a contradiction. Hence $V(H_i) {\setminus } \{h_i\} \subseteq S$, for each $1 \leqslant i \leqslant t$. Therefore we have

$$\begin{aligned} \sum _{i=1}^t|V(H_i)|-t \leqslant \mathrm{cdim}_l(G[H_1,H_2,\dots ,H_k]). \end{aligned}$$

Let $G'$ be the induced subgraph on vertex set $\{h_1,\dots ,h_t,v_{t+1},\dots ,v_k\}$. Clearly $G'$ is isomorphic to G. Now, let $S'$ be an l-clique metric basis for $G'$. Since, for each $h_j,h'_j \in V(H_j)$, where $t+1 \leqslant j \leqslant k$, we have $d(L,h_j)=d(L,h'_j)$, where L is an l-clique, $S' \cup \bigcup _{i=1}^t (V(H_i) {\setminus } \{h_i\})$ is an l-clique metric generator for $G[H_1,H_2,\dots ,H_k]$. So

$$\begin{aligned} \mathrm{cdim}_l(G[H_1,H_2,\dots ,H_k]) \leqslant \mathrm{cdim}_l(G)+\sum _{i=1}^t|V(H_i)|-t. \end{aligned}$$

$\square$

In the following theorem, we determine the l-CMD of the G-generalized join graph $G[H_1,H_2,\dots ,H_n]$, in the case that $H_i$’s are empty graphs and G is a path $P_n$. In fact the following theorem shows examples where the bounds in Proposition 2.8 are reached.

Theorem 2.9

Assume that G is a path on $n\geqslant 2$ vertices with $V(G)=\{v_1,v_2,\dots ,v_n\}$, and let $H_1,H_2,\dots ,H_n$ be n pairwise disjoint empty graphs. Then $\sum _{i=1}^n|V(H_i)|-n \leqslant \mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) \leqslant \sum _{i=1}^n|V(H_i)|-n+1$, when $l \in \{1,2\}$. Also if $|V(H_i)|>1$, for each $1 \leqslant i \leqslant n$, then we have

$$\begin{aligned} \mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) = \left\{ \begin{array}{ll} \sum _{i=1}^n|V(H_i)|-n+1 &{}\quad n=3, \; l=1,2\\ \sum _{i=1}^n|V(H_i)|-n &{}\quad n\ne 3, \; l=1,2\\ 1 &{}\quad l \geqslant 3. \end{array} \right. \end{aligned}$$

Proof

If $l \geqslant 3$, then clearly $\mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) =1$. So let $l \in \{1,2\}$. Let $h_1, \dots , h_n$ be arbitrary vertices in $H_1,\dots ,H_n$, respectively. Set $S=\bigcup _{i=1}^n(V(H_i) {\setminus } \{h_i\})$, where $h_i$ is an arbitrary vertex in $H_i$. By Proposition 2.8, every l-clique metric generator of $G[H_1,H_2,\dots ,H_n]$ contains S. Also $S \cup \{h_1\}$ is an l-clique metric generator for $G[H_1,H_2,\dots ,H_n]$. Hence we have

$$\begin{aligned} \sum _{i=1}^n|V(H_i)|-n \leqslant \mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) \leqslant \sum _{i=1}^n|V(H_i)|-n+1. \end{aligned}$$

If $n=3$, then we have $r_{G[H_1,H_2,H_3]}^1(h_1|S)=r_{G[H_1,H_2,H_3]}^1(h_3|S)$ and also we have $r_{G[H_1,H_2,H_3]}^2(h_1h_2|S)=r_{G[H_1,H_2,H_3]}^2(h_2h_3|S)$, which means that S is not an l-clique metric generator of $G[H_1,H_2,H_3]$, and as a consequence, $\mathrm{cdim}_l(G[H_1,H_2,H_3])> |S|=\sum _{i=1}^3|V(H_i)|-3$. Set $S'=S \cup \{h_1\}$. Now, one can see that $S'$ is an l-clique metric basis of $G[H_1,H_2,H_3]$, and so $\mathrm{cdim}_l(G[H_1,H_2,H_3])=\sum _{i=1}^3|V(H_i)|-2$. Now, let $|V(H_i)|>1$, for each $1 \leqslant i \leqslant n$ and, assume that $n \ne 3$. Then it is easy to see that S is an l-clique metric generator of $G[H_1,H_2,\dots ,H_n]$, which implies that $\mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) =\sum _{i=1}^n|V(H_i)|-n$. $\square$

In the following theorem, we determine the l-CMD of the G-generalized join graph $G[H_1,H_2,\dots ,H_n]$, in the case that $H_i$’s are empty graphs and G is the complete graph $K_n$.

Theorem 2.10

Assume that $G \cong K_n$ with $V(G)=\{v_1,v_2,\dots ,v_n\}$, $n >2$, and let $H_1,H_2,\dots ,H_n$ be n pairwise disjoint empty graphs such that the number of trivial $H_i$’s is $r<n$. Then we have

$$\begin{aligned} \mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) = \left\{ \begin{array}{ll} \sum _{i=1}^n|V(H_i)|-1 &{}\quad 2 \leqslant l \leqslant n-1\\ \sum _{i=1}^n|V(H_i)|-n+r-1 &{}\quad l=1, \; r>0\\ \sum _{i=1}^n|V(H_i)|-n &{}\quad l=1, \; r=0\\ \sum _{i=1}^n|V(H_i)|-n &{}\quad l=n. \end{array} \right. \end{aligned}$$

Proof

Assume that $h_1, \dots , h_n$ are arbitrary vertices in $H_1,\dots ,H_n$, respectively. Let $S=\bigcup _{i=1}^n(V(H_i) {\setminus } \{h_i\})$. By Proposition 2.8, every l-clique metric generator of $G[H_1,H_2,\dots ,H_n]$ contains S, which implies that $\mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) \geqslant \sum _{i=1}^n|V(H_i)|-n$. First assume that $l=1$. Since the places in which there is a 2, if exists, appears in the l-clique metric S-representation of each two distinct $h_i$ and $h_j$, with $1 \leqslant i\ne j \leqslant n$, are different from each other, their l-clique metric S-representations are not equal. Without loss of generality, assume that $|V(H_1)|=\dots =|V(H_r)|=1$. Hence the l-clique metric S-representation of all $h_i$’s, for $1 \leqslant i \leqslant r$ is equal. So, in this situation, any l-clique metric generator of $G[H_1,H_2,\dots ,H_n]$ is of the form $S \cup \bigcup _{i=1,i \ne j}^r\{h_i\}$, for some $1 \leqslant j \leqslant r$. Hence we have $\mathrm{cdim}_1(G[H_1,H_2,\dots , H_n])= \sum _{i=1}^n|V(H_i)|-n+r-1$, for $0<r<n$. Clearly if $r=0$, then S is a 1-clique metric basis of $G[H_1,H_2,\dots ,H_n]$, and so $\mathrm{cdim}_1(G[H_1,H_2,\dots , H_n]) = \sum _{i=1}^n|V(H_i)|-n$.

Now, assume that $l \geqslant 2$. Let $S'$ be an l-clique metric generator and L be an arbitrary l-clique of $G[H_1,H_2,\dots ,H_n]$. For each $x \in S'$, we have

$$\begin{aligned} d_{G[H_1,H_2,\dots , H_n]}(L,x) = \left\{ \begin{array}{ll} 1 &{}\quad x \notin L\\ 0 &{}\quad x \in L. \end{array} \right. \end{aligned}$$

So, for each two distinct l-cliques $L_1$ and $L_2$, $L_1 \cap S'=L_2 \cap S'$ if and only if $L_1$ and $L_2$ have the same l-clique metric $S'$-representations. If $l=n$, then, for each two distinct l-cliques $L_1$ and $L_2$, $L_1 \cap S=L_2 \cap S$ implies that $L_1=L_2$. This implies that S is an l-clique metric basis, and so $\mathrm{cdim}_n(G[H_1,H_2,\dots , H_n])= \sum _{i=1}^n|V(H_i)|-n$. Now, assume that $2 \leqslant l \leqslant n-1$. If there are $h_i$ and $h_j$ with $1 \leqslant i\ne j \leqslant n$ such that they do not belong to an l-clique metric generator $S'$, then consider two l-cliques $L_1$ and $L_2$ with $h_i \in L_1$, $h_j \in L_2$ and $L_1 {\setminus } \{h_i\}=L_2 {\setminus } \{h_j\}$. Since $L_1 \cap S'=L_2 \cap S'$, they have the same l-clique metric $S'$-representations, which is impossible. So in this situation, any l-clique metric generator is of the form $S \cup \bigcup _{i=1, i \ne j}^nV(H_i)$, for some $1 \leqslant j \leqslant n$. Thus we have $\mathrm{cdim}_l(G[H_1,H_2,\dots , H_n])= \sum _{i=1}^n|V(H_i)|-1$. $\square$

In the following theorem, we determine the l-CMD of the G-generalized join graph $G[H_1,H_2,\dots ,H_n]$, in the case that $H_i$’s are empty graphs and G is isomorphic to the cycle $C_n$, where $n>3$. Note that the case $n=3$ is obtained by Theorem 3.3.

Theorem 2.11

Assume that G is a cycle $C_n$ with vertex set $V(G)=\{v_1,v_2,\dots ,v_n\}$, $n>3$, and let $H_1,H_2,\dots ,H_n$ be n pairwise disjoint empty graphs. Then

$$\begin{aligned} \sum _{i=1}^n|V(H_i)|-n \leqslant \mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) \leqslant \sum _{i=1}^n|V(H_i)|-n+2, \end{aligned}$$

when $l \in \{1,2\}$, and $\mathrm{cdim}_l(G[H_1,H_2,\dots , H_n])=1$, for $l \geqslant 3$. Also, for $n=4$ and $l \in \{1,2\}$, we have $\mathrm{cdim}_l(G[H_1,H_2,\dots , H_n])=\sum _{i=1}^n|V(H_i)|-n+2$, and if $|V(H_i)|>1$, for each $1 \leqslant i \leqslant n$, then we have $\mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) =\sum _{i=1}^n|V(H_i)|-n$, when $n> 4$ and $l \in \{1,2\}$.

Proof

Clearly if $l \geqslant 3$, then $\mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) =1$. So assume that $l \in \{1,2\}$. Let $h_1, \dots , h_n$ be arbitrary vertices in $H_1,\dots ,H_n$, respectively, and $S=\bigcup _{i=1}^n(V(H_i) {\setminus } \{h_i\})$. By Proposition 2.8, every l-clique metric generator of $G[H_1,H_2,\dots ,H_n]$, contains S. Also $S\cup \{h_1,h_2\}$ is an l-clique metric generator of $G[H_1,H_2,\dots ,H_n]$. Hence $\sum _{i=1}^n|V(H_i)|-n \leqslant \mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) \leqslant \sum _{i=1}^n|V(H_i)|-n+2$. If $n=4$, then one can see that $S\cup \{h_1,h_2\}$ is an l-clique metric basis of $G[H_1,H_2,\dots ,H_n]$. So $\mathrm{cdim}_l(G[H_1,H_2,H_3, H_4]) =\sum _{i=1}^4|V(H_i)|-2$.

Now, assume that $n \geqslant 5$. Let $|V(H_i)|>1$, for each $1 \leqslant i \leqslant n$. Since $n \geqslant 5$ and $|V(H_i)| \geqslant 2$, for any two vertices $h_i,h_j \notin S$, the distance between $h_i$ and any vertex belonging to $S \cap (V(H_{i-1})\cup V(H_{i+1}))$ is one, while the distance between $h_j$ and any vertex belonging to at least one of these two sets $S\cap V(H_{i-1})$ or $S \cap V(H_{i+1})$ is different than one. Thus, S is an 1-clique metric generator for $G[H_1,H_2,\dots , H_n]$. Now, let $L_1$ and $L_2$ be two distinct 2-cliques. If $L_1 \cap S=\phi =L_2 \cap S$, then the places that 1 appears in their 2-clique metric S-representations are different. So, without loss of generality, assume that $s \in L_1 \cap S$. If $s \notin L_2$, then the corresponding components to s in the 2-clique metric S-representations of $L_1$ and $L_2$ are zero and nonzero, respectively. Thus, let $s \in L_2$. If $L_1 \subseteq S$ or $L_2 \subseteq S$, then clearly their 2-clique metric S-representations are different. Now, assume that $L_1 \nsubseteq S$ and $L_2 \nsubseteq S$. Then one can see that the places of 1 in their 2-clique metric S-representations are different. So S is an 2-clique metric generator for $G[H_1,H_2,\dots , H_n]$. Hence we have $\mathrm{cdim}_l(G[H_1,H_2,\dots , H_n]) = \sum _{i=1}^n|V(H_i)|-n$. $\square$

3 l-Clique Metric Dimension of $\Gamma ({\mathbb {Z}}_n)$

Let R be a commutative ring with nonzero identity. We denote the set of all unit elements and zero divisors of R by U(R) and Z(R), respectively. Also by $Z^*(R)$ we denote the set $Z(R) {\setminus } \{0\}$. Sharma and Bhatwadekar [20] defined the comaximal graph of a commutative ring R. The comaximal graph of R is a simple graph whose vertices consists of all elements of R, and two distinct vertices a and b are adjacent if and only if $aR+bR=R$, where cR is the ideal generated by c, for $c \in R$. Let $\Gamma (R)$ be an induced subgraph of the comaximal graph with nonunit elements of R as vertices. The properties of the graph $\Gamma (R)$ were studied in [16, 22, 25].

For two integers r and s, the notation (r, s) stands for the greatest common divisor of r and s. Also we denote the elements of the ring ${\mathbb {Z}}_n$, where $n>1$, by $0,1,2, \dots ,n-1$. For every nonzero element a in ${\mathbb {Z}}_n$, if $(a,n)=1$, then a is a unit element; otherwise, $(a,n)\ne 1$, and so a is a zerodivisor. Therefore, $|U({\mathbb {Z}}_n)|=\phi (n)$ and $|Z({\mathbb {Z}}_n)|=n - \phi (n)$, where $\phi$ is the Euler’s totient function.

An integer d is said to be a proper divisor of n if $1<d<n$ and $d \mid n$. Now let $d_1,d_2, \dots ,d_k$ be the distinct proper divisors of n. For $1 \leqslant i \leqslant k$, set

$$\begin{aligned} A_{d_i}:=\{x \in {\mathbb {Z}}_n \mid (x,n)=d_i\}. \end{aligned}$$

Clearly, the sets $A_{d_1}, A_{d_2}, \dots ,A_{d_k}$ are pairwise disjoint and we have

$$\begin{aligned} Z^*({\mathbb {Z}}_n)=A_{d_1}\cup A_{d_2}\cup \dots \cup A_{d_k} \end{aligned}$$

and

$$\begin{aligned} V(\Gamma ({\mathbb {Z}}_n))=\{0\} \cup A_{d_1}\cup A_{d_2}\cup \dots \cup A_{d_k}. \end{aligned}$$

The following lemma is stated from [27].

Lemma 3.1

[27,  Proposition 2.1] Let $1\leqslant i \leqslant k$. Then $|A_{d_i}|=\phi (\frac{n}{d_i})$.

In this section, the induced subgraph of $\Gamma ({\mathbb {Z}}_n)$ on the set $A_{d_i}$ is denoted by $\Gamma (A_{d_i})$, where $1\leqslant i \leqslant k$.

The following lemma states some adjacencies in $\Gamma ({\mathbb {Z}}_n)$.

Lemma 3.2

The following statements hold:

• (i) Two distinct vertices x and y are adjacent in $\Gamma ({\mathbb {Z}}_n)$ if and only if $(x,y) \in U({\mathbb {Z}}_n)$.

• (ii) For $1\leqslant i \leqslant k$, $\Gamma (A_{d_i})$ is isomorphic to ${\overline{K}}_{\phi (\frac{n}{d_i})}$.

• (iii) For $1\leqslant i\ne j \leqslant k$, a vertex of $A_{d_i}$ is adjacent to a vertex of $A_{d_j}$ if and only if $(d_i,d_j)=1$.

Proof

• (i) First suppose that x and y are adjacent vertices in $\Gamma ({\mathbb {Z}}_n)$. Assume on the contrary that $d=(x,y)\notin U({\mathbb {Z}}_n)$. So we have $x{\mathbb {Z}}_n \subseteq d{\mathbb {Z}}_n$ and $y{\mathbb {Z}}_n \subseteq d{\mathbb {Z}}_n$. Thus $x{\mathbb {Z}}_n + y{\mathbb {Z}}_n \subseteq d{\mathbb {Z}}_n \ne {\mathbb {Z}}_n$, and this means that x and y are not adjacent, which is a contradiction. Now, let $u=(x,y)\in U({\mathbb {Z}}_n)$. So there exist $r,s \in {\mathbb {Z}}$ such that $u=rx+sy \in x{\mathbb {Z}}_n+y{\mathbb {Z}}_n$. Therefore we have $x{\mathbb {Z}}_n+y{\mathbb {Z}}_n={\mathbb {Z}}_n$, which implies that x and y are adjacent.

• (ii) For each two distinct elements $x,y \in A_{d_i}$, we have $(x,n)=d_i=(y,n)$. So $d_i \mid (x,y)$, which implies that $(x,y)\notin U({\mathbb {Z}}_n)$. Hence by (i), we have that x and y are not adjacent. Therefore by Lemma 3.1, we have $\Gamma (A_{d_i}) \cong {\overline{K}}_{\phi (\frac{n}{d_i})}$.

• (iii) Let $i,j \in \{1,2,\dots ,k\}$ with $i \ne j$. First assume that $x\in A_{d_i}$ and $y \in A_{d_j}$ are adjacent vertices. If $(d_i,d_j)=d\ne 1$, then $(n,d)=d$. Since $(x,n)=d_i$ and $(y,n)=d_j$, we have that $d \mid x,y$. Hence $Rx+Ry \subseteq Rd \ne R$, which is impossible. Now suppose that $(d_i,d_j)=1$. Let $x\in A_{d_i}$ and $y\in A_{d_j}$ be arbitrary vertices. If $d=(x,y)\notin U({\mathbb {Z}}_n)$, then $t=(d,n)\ne 1$. Since $t \mid x,y,n$, we have $t \mid (d_i,d_j)$ and this is impossible. Hence $(x,y)\in U({\mathbb {Z}}_n)$ which means that x and y are adjacent. $\square$

Now, we introduce a simple graph $G_n$, which plays an important role in the structure of $\Gamma ({\mathbb {Z}}_n)$. The graph $G_n$ is the simple graph with vertex set $\{d_1,d_2,\dots ,d_k\}$, where $d_i$’s, $1\leqslant i \leqslant k$, are the proper divisors of n, and two distinct vertices $d_i$ and $d_j$ are adjacent if and only if $(d_i,d_j)=1$.

Let $n=p_1^{\alpha _1}p_2^{\alpha _2}\dots p_t^{\alpha _t}$ be the factorization of n to its prime powers, where $t,\alpha _1,\dots ,\alpha _t$ are positive integers and $p_1,\dots ,p_t$ are distinct prime numbers. Every divisor of n is of the form $p_1^{\beta _1}p_2^{\beta _2}\dots p_t^{\beta _t}$, for some integers $\beta _1,\dots ,\beta _t$, where $0 \leqslant \beta _i \leqslant \alpha _i$ for each $i\in \{1,2,\dots ,t\}$. Hence the number of proper divisors of n is equal to $\prod _{i=1}^t(n_i+1)-2$. Therefore we have $k=|V(G_n)|=\prod _{i=1}^t(n_i+1)-2$.

Let $\Gamma ^*({\mathbb {Z}}_n)=\Gamma ({\mathbb {Z}}_n) {\setminus } \{0\}$. Consider the graph $G_n$ and replace each vertex $d_i$ of $G_n$ by $\Gamma [A_{d_i}]$. In view of Lemma 3.1, we have

$$\begin{aligned} \Gamma ^*({\mathbb {Z}}_n)=G_n\left[ {\overline{K}}_{\phi \left( \frac{n}{d_1}\right) }, {\overline{K}}_{\phi \left( \frac{n}{d_2}\right) },\dots ,{\overline{K}}_{\phi \left( \frac{n}{d_k}\right) }\right] . \end{aligned}$$

Now, since the zero element is adjacent to none of the vertices of $\Gamma ^*({\mathbb {Z}}_n)$, we have

$$\begin{aligned} \Gamma ({\mathbb {Z}}_n)=(K_1\cup \Gamma ^*({\mathbb {Z}}_n)). \end{aligned}$$

In the following theorem, we study the l-CMD of $\Gamma ({\mathbb {Z}}_n)$.

Theorem 3.3

Assume that $\{d_1,d_2,\dots ,d_t\}$, where $1\leqslant t \leqslant k$, are those vertices of $G_n$ that each of them belongs to an l-clique. Then for $l=1$ we have

$$\begin{aligned} \sum _{i=1}^k \phi \left( \frac{n}{d_i}\right) -k+r \leqslant \mathrm{cdim}_l(\Gamma ({\mathbb {Z}}_n)) \leqslant \mathrm{cdim}_l(G_n)+\sum _{i=1}^k \phi \left( \frac{n}{d_i}\right) -k+r \end{aligned}$$

and for $l>1$,

$$\begin{aligned} \sum _{i=1}^t \phi \left( \frac{n}{d_i}\right) -t \leqslant \mathrm{cdim}_l(\Gamma ({\mathbb {Z}}_n)) \leqslant \mathrm{cdim}_l(G_n)+\sum _{i=1}^t \phi \left( \frac{n}{d_i}\right) -t, \end{aligned}$$

where r is the number of isolated vertices of $G_n$.

Proof

Note that the graph $G_n$ is not connected in general. Let r be the number of isolated vertices of $G_n$. Since 0 is the isolated vertex of $\Gamma ({\mathbb {Z}}_n)$, we assume that $0,a_1, \dots , a_r$ are the isolated vertices of $\Gamma ({\mathbb {Z}}_n)$. By Remark 2.7, we have

$$\begin{aligned} \mathrm{cdim}_1(\Gamma ({\mathbb {Z}}_n))=\mathrm{cdim}_1(\Gamma ({\mathbb {Z}}_n) {\setminus } \{0,a_1, \dots , a_r\}) +r. \end{aligned}$$

Now, the results follow from Proposition 2.8 and Remark 2.7. $\square$

Example 3.4

Consider the ring ${\mathbb {Z}}_{12}$. We have $d_1=2, d_2=3,d_3=4$, and $d_4=6$. Then $G_{12}$ is the graph $2\sim 3 \sim 4 \cup \{6\}$, which is isomorphic to $P_3 \cup K_1$. Hence we have

$$\begin{aligned} \Gamma ({\mathbb {Z}}_{12})=K_1 \cup G_{12}[{\overline{K}}_2, {\overline{K}}_2,{\overline{K}}_2,K_1] \end{aligned}$$

and, by Theorems 2.9 and 3.3 , we have

$$\begin{aligned} \mathrm{cdim}_l(\Gamma ({\mathbb {Z}}_{12})) =\left\{ \begin{array}{ll} 5 &{}\quad l=1 \\ 4 &{}\quad l=2 \\ 1 &{}\quad l \geqslant 3. \end{array} \right. \end{aligned}$$

In the rest of this section, we discuss the CMD of $\Gamma ({\mathbb {Z}}_n)$, for (i) $n=p^t$, (ii) $n=pq$ and (iii) $n=p^2q$, where p and q are distinct prime numbers and t is a positive integer.

• (i) Let $n=p^t$. Then $\Gamma ({\mathbb {Z}}_{p^t})$ is an empty graph with $p^t - \phi (p^t)=p^{t-1}$ vertices, and so $\Gamma ({\mathbb {Z}}_{p^t})=\overline{K_{p^{t-1}}}$. Now, by Remark 2.7 we have

  $$\begin{aligned} \mathrm{cdim}_l(\Gamma ({\mathbb {Z}}_{p^t})) =\left\{ \begin{array}{ll} p^{t-1}-1 &{}\quad l=1 \\ 1 &{}\quad l \geqslant 2. \end{array} \right. \end{aligned}$$

• (ii) Let $n=pq$, where p and q are distinct prime numbers. Since the only proper divisors of n are p and q, the graph $G_{pq}$ is $p \sim q$. So we have

  $$\begin{aligned} \Gamma ({\mathbb {Z}}_{pq})=K_1\cup G_{pq}[{\overline{K}}_{\phi (q)},{\overline{K}}_{\phi (p)}]. \end{aligned}$$

Now, by Theorem 2.9, we have

$$\begin{aligned} \mathrm{cdim}_l(\Gamma ({\mathbb {Z}}_{pq})) =\left\{ \begin{array}{ll} p+q-4 &{}\quad l=1,2 \\ 1 &{}\quad l \geqslant 3. \end{array} \right. \end{aligned}$$

• (iii) Let $n=p^2q$, where p and q are distinct prime numbers. Since p, q, and pq are the proper divisors of n, the graph $G_{p^2q}$ is $p \sim q \sim p^2 \cup \{pq\}$. Hence we have

  $$\begin{aligned} \Gamma ({\mathbb {Z}}_{p^2q})=K_1\cup G_{p^2q}[{\overline{K}}_{\phi (pq)}, {\overline{K}}_{\phi (p^2)},{\overline{K}}_{\phi (q)},{\overline{K}}_{\phi (p)}]. \end{aligned}$$

Since $\phi (pq)=pq-p-q+1$ and $\phi (p^2)=p^2-p$, by Theorem 2.9 and Remark 2.7,

$$\begin{aligned} \mathrm{cdim}_l(\Gamma ({\mathbb {Z}}_{p^2q})) =\left\{ \begin{array}{ll} p^{2}+pq-p-3 &{}\quad l=1 \\ p^{2}+pq-2p-2 &{}\quad l=2 \\ 1 &{}\quad l \geqslant 3. \end{array} \right. \end{aligned}$$

4 l-Clique Metric Dimension Over Corona Product

Let G and H be two graphs with the vertex sets $\{g_1,\ldots , g_n\}$ and $\{h_1,\ldots ,h_m\}$, respectively. The corona of G and H, denoted by $G\circ H$, is the graph whose vertex and edge sets are defined as below:

$$ \begin{aligned} V(G\circ H)&=V(G)\cup (\cup _{i=1}^n\{h_{1_i},\ldots ,h_{m_i}\}),\\ E(G\circ H)&=E(G)\cup \{h_{j_i}h_{l_i} :h_jh_l\in E(H)\, \& \,1\le i\le n\}\\&\qquad \cup \{g_ih_{j_i} : 1\le j\le m, \, 1\le i\le n\}. \end{aligned}$$

The metric dimension (1-CMD) of corona product graphs was investigated in [26]. After that Peterin and Yero studied the edge metric dimension (2-CMD) over corona product in [18]. In this section, we give a formula for the l-CMD of corona product of two graphs G and H for $l\ge 3$. In what follows, we say the vertex v distinguishes two l-cliques U and W if $d(v,U)\ne d(v,W)$.

Theorem 4.1

Let G and H be two connected graphs of order n and m, respectively, and $l\ge 3$ be an integer number. If $\{V_1(H),\ldots ,V_k(H)\}$ is the $(l-1)$-clique set of H, then

$$\begin{aligned}\mathrm{cdim}_l(G\circ H)={\left\{ \begin{array}{ll} \mathrm{cdim}_l(G) &{} \mathrm{if}\; \omega (H)<l-1\\ \mathrm{dim}(G) &{} \mathrm{if} \; k=1 \; \mathrm{and}\; \omega (G)<l \end{array}\right. }, \end{aligned}$$

where $\omega (G)$ and $\omega (H)$ are the clique numbers of G and H, respectively.

Proof

Let $V(G)=\{g_1, \ldots , g_n\}$ and $H_i$ be the i-th copy of H in $G\circ H$, $1\le i\le n$. Then $G\circ H$ is obtained by joining each vertex of the i-copy of H to the i-th vertex, $g_i$, of G.

Let $S_G$ be an l-clique metric basis of G and $\{V_1(G),\ldots , V_t(G)\}$ be the l-clique set of G. Also, let $V_{j_i}(H)$ denote the i-the copy of $V_j(H)$ in $G\circ H$, for $1\le i\le n$ and $1\le j\le k$. Thus, it is clear that $V'_{j_i}(H)=V_{j_i}(H)\cup \{g_i\}$, $1\le i\le n$, is an l-clique in $G\circ H$.

First, we prove that if $\omega (H)<l-1$ (or $k=0$), then $\mathrm{cdim}_l(G\circ H)=\mathrm{cdim}_l(G)$. To do this, we prove that $S_G$ is also an l-clique metric basis of $G\circ H$. Clearly $S_G$ is an l-clique metric generator for $G\circ H$ and so $\mathrm{cdim}_l(G\circ H)\le \mathrm{cdim}_l(G)$. Suppose that S is an l-clique metric basis of $G\circ H$. We claim that $|S\cap V(H_i)|\le 1$ for $1\le i\le n$. To prove this claim, suppose, on the contrary that there exist $u,z\in S\cap V(H_i)$. Then $S'=S{\setminus } \{u\}$ is not an l-clique metric generator for $G\circ H$. Thus there exist two l-cliques U and W in $G\circ H$ such that $d_{G\circ H}(v,U)=d_{G\circ H}(v,W)$ for each $v\in S'$. Hence $d_{G\circ H}(z,U)=d_{G\circ H}(z,W)$. On the other hand, since $\omega (H)<l-1$, then $d_{G\circ H}(z,U)=d_{G\circ H}(z,W)=d_G(g_i,U)+1=d_G(g_i,W)+1$. Also, since $\omega (H)<l-1$, then $d_{G\circ H}(u,U)=d_{G\circ H}(u,W)=d_G(g_i,U)+1=d_G(g_i,W)+1$. Therefore S is not an l-clique metric generator for $G\circ H$ which is a contradiction.

Now suppose that $u\in S\cap V(H_i)$. Then $S'=(S-\{u\})\cup \{g_i\}$ is also an l-clique metric basis of $G\circ H$. Because $d_{G\circ H}(u, V_j(G))=d_G(g_i,V_j(G))+1$ for each $1\le j\le t$. By repeating this technique, we reach an l-clique metric basis $S''$ of $G\circ H$ with this property that all vertices of $S''$ are in G. Therefore, $\mathrm{cdim}_l(G\circ H)\ge \mathrm{cdim}_l(G)$.

Now, suppose that $\omega (G)<l$, $k=1$ and $V_1(H)$ is the $(l-1)$-clique of H. Let $S_G$ be a 1-clique metric basis of G. We claim that $S_G$ is an l-clique metric generator for $G\circ H$. Then, since $d_{G\circ H}(V_{1_i}'(H),v)=d_G(g_i,v)$ for each $v\in S_G$, then every pair of l-cliques $V_{1_i}(H)$’s, $1\le i\le n$, is distinguished by a vertex of $S_G$. Therefore, $S_G$ is an l-clique metric generator for $G\circ H$ and so $\mathrm{cdim}_l(G\circ H)\le |S_G|=\mathrm{dim}(G)$. Then, it is sufficient to show that $\mathrm{cdim}_l(G\circ H)\ge \mathrm{dim}(G)$. To do this, suppose that $S'$ is an l-clique metric basis of $G\circ H$. By the above argument, if $|S'\cap V(G)|=|S'|$, then we have nothing to prove. Otherwise, there exists $v\in S'$ such that $v\in V_{1_i}$ for an $i\in \{1,\ldots , n\}$. Since $d_{G}(v,V'_{1_j})=d_{G}(g_i,V'_{1_j})+1$ for $i\ne j\in \{1,\ldots , n\}$, then $S''=(S-v)\cup \{g_i\}$ is also an l-clique metric basis of $G\circ H$. We use this technique to reach an l-clique metric basis $S'''$ of $G\circ H$ with this property that $|S'''\cap V(G)|=|S'''|$. Therefore, $\mathrm{cdim}_l(G\circ H)\ge \mathrm{dim}(G)$. $\square$

The concept of global forcing sets for maximal matchings was presented in [24]. Here we need to introduce an extension of the idea of global forcing sets for l-cliques of a graph.

A global forcing set for l-cliques of a graph G is a subset S of V(G) with this property that $V_1\cap S\ne V_2\cap S$ for any two l-cliques $V_1$ and $V_2$ of G. A global forcing set for l-cliques of G with minimum cardinality is called a minimum global forcing set for l-cliques of G, and its cardinality, denoted by $\varphi _l$, is the global forcing number for l-cliques of G.

We can find a global forcing set for l-cliques of G by the following ILP.

Let G be a graph with $V(G)=\{v_1, \ldots v_n\}$ and let $\{V_1, \ldots , V_k\}$ be the set of all l-cliques of G. Let $D_G=[d_{ij}]$ be a $k\times n$ matrix, where $d_{ij}= 1$ if $v_j\in V_i$, and $d_{ij}= 0$ otherwise. Let $F: \{0,1\}^{n} \rightarrow {{\mathbb {N}}}_0$ be defined by

$$\begin{aligned} F(x_1,\ldots , x_{n}) = x_1 + \cdots + x_n. \end{aligned}$$

Then our goal is to determine $\min F$ subject to the constraints

$$\begin{aligned} |d_{i1}-d_{j1}|x_1+|d_{i2}-d_{j2}|x_2 + \cdots + |d_{in}-d_{jn}|x_n>0,\quad 1 \le i < j \le k. \end{aligned}$$

Note that if $x'_1, \ldots , x'_n$ is a set of values for which F attains its minimum, then $S = \{v_i: x'_i =1\}$ is a minimum global forcing set for l-cliques of G.

Theorem 4.2

Let G and H be two connected graphs with $|V(G)|=n$, and $l\ge 3$ be an integer number. If $\{V_1(H),\ldots ,V_k(H)\}$ is the $(l-1)$-clique set of H and $\omega (H)=l-1$, then for $k\ge 2$ we have

$$\begin{aligned} \mathrm{cdim}_l(G\circ H)= n\cdot \varphi _{l-1}(H) . \end{aligned}$$

Proof

Let S be an l-clique metric generator for $G\circ H$. Suppose, on the contrary that there exists $H_i$, a copy of H in $G\circ H$, that $|S\cap V(H_i)|<\varphi _{l-1}(H)$. Then there exist two $(l-1)$-cliques $V_{j_i}(H)$ and $V_{q_i}(H)$ in $H_i$ such that $S\cap V_{j_i}(H)=S\cap V_{q_i}(H)$. Hence $d_{G\circ H}(u, V_{j_i}(H))=d_{G\circ H}(u, V_{q_i}(H))=0$ for each $u\in S\cap V_{j_i}(H)$, and $d_{G\circ H}(u, V_{j_i}(H))=d_{G\circ H}(u, V_{q_i}(H))=1$ for each $u\in S\cap (V(H_i){\setminus } V_{j_i}(H))$. On the other hand, it is not difficult to check that $d_{G\circ H}(u, V_{j_i}(H))=d_{G\circ H}(u, V_{q_i}(H))$ for each $u\in S{\setminus } V(H_i)$. Thus, $d_{G\circ H}(u,V_{j_i}(H))=d_{G\circ H}(u,V_{q_i}(H))$ for each $u\in S$, which is contrary to our assumption. Therefore, $\mathrm{cdim}_l(G\circ H)\ge n\cdot \varphi _{l-1}(H)$.

It remains to prove that $\mathrm{cdim}_l(G\circ H)\le n\cdot \varphi _{l-1}(H)$. Let $S_H$ be a minimum global forcing set for ($l-1$)-cliques of H, and let $S_{H_i}$ be the i-th copy of $S_H$ in $G\circ H$. Then, it is easy to check that $S'=\bigcup _{i=1}^nS_{H_i}$ is an l-clique metric generator for $G\circ H$. Therefore, $\mathrm{cdim}_l(G\circ H)\le n\cdot \varphi _{l-1}(H)$. $\square$

5 Complexity Issues

The clique problem is the optimization problem of finding a clique of maximum size in a graph. As a decision problem, we ask simply whether a clique of a given size k exists in the graph.

Theorem 5.1

[8] The clique problem is $\textit{NP}$-complete.

Therefore, the problem of finding all l-cliques in a graph is NP-hard. Hence, throughout this section we are assuming that all the l-cliques of the graph are given.

In this section, we prove the l-CMD problem is NP-complete. Recall that for $l=1,2$, l-CMD problems are the metric dimension and the edge metric dimension problems, respectively. On the other hand, Garey and Johnson [11] proved that the decision version of the metric dimension problem is NP-complete on connected graphs. Also, NP-completeness of computing the edge metric dimension of connected graphs was proved in [14]. Moreover, Epstein, Levin, and Woeginger showed that for split graphs, bipartite graphs, co-bipartite graphs, and line graphs of bipartite graphs, the problem of computing the metric dimension of the graph is NP-hard [10]. Then, we prove NP-completeness of computing the l-CMD of connected graphs for $L\ge 3$. Let us start with the below decision problem.

l-CMD problem: For a given positive integer l. Let G be a connected graph with n where $n \ge 3$, X be the set of all distinct l-cliques of G, and let r be a positive integer such that $1 \le r \le n-1$. Is $\mathrm{cdim}_l(G) \le r$?

Note that the l-CMD problem is the decision version of the problem of computing $\mathrm{cdim}_l(G)$ for a given connected graph G.

Our proof for showing that the NP-completeness of l-CMD problem is based on a reduction from the metric dimension problem on connected bipartite graphs. We recommend [7] for more details on the reduction technique. Now, we are ready to prove that the l-CMD problem is NP-complete.

Theorem 5.2

The l-CMD problem, for $l\ge 3$, is NP-complete.

Proof

Note that the l-CMD problem is clearly in NP because we can check its feasibility as a l-clique metric generator in polynomial time.

For showing NP-hardness of this problem, we present a reduction from the metric dimension for connected bipartite graphs.

Let G be a connected bipartite graph where $V(G)=\{g_1,\ldots ,g_n\}$. Now, we construct graph $G'$ from G by taking one copy of G and n copies of the complete graph $K_{l-1}$ and by joining each vertex of the i-th copy of $K_{l-1}$ to the i-th vertex of G, $i=1,\ldots ,n$. In other words, $G'=G\circ K_{l-1}$. For more illustration, see an example of G and $G'$ in Fig. 1. Since G is bipartite, then $\omega (G)<3$. Thus by Theorem 4.1, $\mathrm{cdim}_l(G')=\mathrm{cdim}_l(G\circ H)=\mathrm{dim}(G)$. Moreover, it is easy to see that constructing $G'$ from G can be done in polynomial time. Therefore, if there exists a polynomial-time algorithm for computing $\mathrm{cdim}_l(G')$, then there exists a polynomial-time algorithm for computing $\mathrm{dim}(G)$. $\square$

Fig. 1

The graph $G'$ constructed from G for $l=3$

An integer linear programming (ILP) model for the classical metric dimension problem was presented in [6]. Motivated by this work and using its notations, we consider here an IPL model for computing ${cdim}_l(G)$ for a given connected graph G and its l-cliques. Let $G=(V,E)$ be a connected graphs with $V=\{u_1,\ldots , u_n\}$. Let $V_1,\ldots , V_k$ be the l-cliques of G. Also, suppose that $D_G=[d_{ij}]$ is a $k\times n$ matrix such that $d_{ij}=d_G(V_i,u_j)$ for $i\in \{1,\ldots , k\}$ and $j\in \{1,\ldots , n\}$. Consider the binary decision variables $x_i$ for $i\in \{1,\ldots , n\}$ where $x_i\in \{0,1\}$. By $x_i$, we mean the vertex $u_i$ is a member of an l-clique metric generator of G and $x_i=0$ for otherwise. we define the objective function F by

$$\begin{aligned} F(x_1,\ldots ,x_n)=x_1+\cdots +x_n. \end{aligned}$$

Minimize F subject to the following constraints

$$\begin{aligned} |d_{i1}-d_{j1}|x_1+|d_{i2}-d_{j2}|x_2+\cdots +|d_{in}-d_{jn}|x_n>0,\quad 1\le i<j\le k \end{aligned}$$

is equivalent to finding a basis in the sense that if $x_1',\ldots ,x_n'$ is a set of values for which F attains its minimum, then $W=\{u_i\; | \; x_i'=1\}$ is a basis for G.

Fig. 2

Graph G

For example, consider graph G shown in Fig. 2 with 3-cliques $V_1=\{u_1,u_2,u_3\}$ and $V_2=\{u_3,u_4,u_5\}$. Then, $D_G=\begin{pmatrix} 0 &{} 0&{} 0&{} 1&{} 1\\ 1 &{} 1 &{} 0&{} 0&{} 0 \end{pmatrix}$. Therefore, minimize $F(x_1,x_2,x_3)=x_1+x_2+x_3+x_4+x_5$ subject to the constraints $x_1+x_2+x_4+x_5 > 0$, $x_1, x_2, x_3, x_4, x_5 \in \{0, 1\}$. Thus F attains its minimum for $x_1=1$, $x_2=x_3=x_4=x_5=0$, hence $W=\{u_1\}$ is a 3-clique metric basis for G.