On the edge metric dimension of graphs

1.   Introduction

Throughout this paper, all graphs considered are finite, simple and undirected. We follow the notation and terminology of Bondy ^[1] and Diestel ^[2]. A generator of a metric space is a set $S$ of points in the space with the property that every point of the space is uniquely determined by its distances from the elements of $S$. Nowadays there exist some different kinds of metric generators in graphs (or networks), each one of them studied in theoretical and applied ways, according to its popularity or to its applications. Nevertheless, there exist quite a lot of other points of view which are still not completely taken into account while describing a graph throughout these metric generators. Inspired by this, Kelenc et al. ^[4] proposed the concept of edge metric generator of a graph. Since then, there are some results about it, the details refer to ^[3,5,7].

Let $G = (V, E)$ be a graph. For every vertex $v \in V$, the open neighborhood of $v$ is $N(v) = \{u \in V : uv \in E\}$ and the closed neighborhood of $v$ is $N[v] = N(v) \cup \{v\}$. The degree of a vertex $v \in V$ is defined as $d_G(v) = |N(v)|$. The minimum and maximum degree of a graph $G$ are denoted by $\delta = \delta(G)$ and $\Delta = \Delta(G)$, respectively. The distance between the vertex $w\in V$ and the subetaaph $H\subseteq G$ is defined as $d_G(w, \; H) = \min\{d_G(w, \; v)|v\in V(H)\}$. Particularly, when $H$ is an edge $e = uv\in E$, then $d_G(w, e) = \min\{d_G(w, u), d_G(w, v)\}$. A vertex $w\in V$ distinguishes two edges $e_1, e_2\in E$ if $d_G(e_1, w)\neq d_G(e_2, w)$. We say $S \subseteq V(G)$ an edge metric generator of $G$ if every pair of edges in $E$ can be distinguished by some element of $S$. The edge metric dimension $edim(G)$ of $G$ is the smallest cardinality of an edge metric generator in $G$. An edge metric basis for $G$ is an edge metric generator of $G$ with cardinality $edim(G)$. For any connected graph $G$ of order $n$, $1\leq edim(G)\leq n-1$ are natural bounds on the edge metric dimension. In ^[4], Kelenc et al. proved that $edim(G) = n-2$ when $G$ is a wheel graph of order $n\geq 6$ or a fan graph of order $n\geq 5$. And further, they also gave the lower bound of edge metric dimension $edim(G)\geq n-2$ for any connected graph $G$ with $\Delta = n-1$, where $n$ is the order of $G$. And there is no more results for the graph with edge metric dimension of a given other value. So it is interesting to characterize graphs with large edge metric dimension such as $edim = n-1$ and $edim = n-2$. Recently, Zubrilina ^[6] classified the graphs on $n$ vertices for which $edim(G) = n-1$. They also proposed an open problem as follows.

Problem 1. For which graphs $G$ of order $n$ is $edim(G) = n-2$?

In this paper, we consider the first step for this problem on connected bipartite graphs. And then we classify the class of graphs of $\Delta = n-1$ with $edim = n-2$.

In ^[8], Zhu et al. constructed connected graphs $G$ of order $n$ such that $edim(G) = n-2$ and $edim(G-e) = n-1$, where $e \in E(G)$. This implies that for any subetaaph $H \subseteq G$, the different of the edge metric dimension between the subetaaph $H$ and $G$ is not clear. Therefore, we first investigate the relation between $edim(H)$ and $edim(G)$ for any induced subetaaph $H\subseteq G$. And further, we also consider the relation between $edim(G)$ and clique number $\omega(G)$ of a graph $G$.

To end this section, we list some known results which will be used in the sequel.

Proposition 1.1. ^[4] For any complete bipartite graph $K_{r, t}$ different from $K_{1, 1}$, $edim(K_{r, t}) = dim(K_{r, t}) = r+t-2$.

Proposition 1.2. ^[4] Let $G$ be a connected graph of order $n$. If there is a vertex $v\in V(G)$ of degree $n-1$, then either $edim(G) = n-1$ or $edim(G) = n-2$.

Proposition 1.3. ^[4] Let $G$ be a connected graph of order $n$. If there are two vertices $u, v\in V(G)$ of degree $n-1$, then $edim(G) = n-1$.

2.   Graphs with edim=n$-$2

Given two graphs $G$ and $H$, the join, denoted by $G\vee H$, is the graph obtained from $G$ and $H$ by adding all the possible edges between a vertex of $G$ and a vertex of $H$. The wheel graph $W_{1, n}$ is isomorphic to $C_n\vee K_1$, the fan graph $F_{1, n}$ is isomorphic to $P_n\vee K_1$ and the star graph $S_{1, n}$ is isomorphic to $\overline{K_n}\vee K_1$. The vertex of degree $n$ in the star graph $S_{1, n}$ is denoted as its center vertex.

Proposition 2.1. Let $W_{1, n-1} = C_{n-1}\vee K_1$ be a wheel graph and $G$ be a graph obtained from $W_{1, n-1}$ by deleting any edges on the cycle $C_{n-1}$, where $n\geq 6$. Then $edim(G) = n-2$.

Proof. Suppose $n\geq 6$ and $V(W_{1, n-1}) = \{x, v_1, v_2, \dots, v_{n-2}, v_{n-1}\}$, where the vertex $x$ has degree $n-1$ and the vertices $v_1, v_2, \dots, v_{n-2}, v_{n-1}$ induce the cycle $C_{n-1}$.

Let $S$ be an edge metric generator of $G$. If there exist $v_i, v_j\in (V(C_{n-1})\setminus S)$, then the edges $xv_i$ and $xv_j$ can not be distinguished by any vertex $w\in V(G)\setminus\{v_i, v_j\}$ because $d_G(xv_i, w) = d_G(xv_j, w) = 1$ or $d_G(xv_i, w) = d_G(xv_j, w) = 0$. So $|V(C_{n-1})\setminus S|\leq 1$ and then $|S|\geq n-2$. From the arbitrariness of $S$, we have $edim(G)\geq n-2$.

Next, it suffices to show that $edim(G)\leq n-2$. Set $S_0 = \{v_1, v_2, \dots, v_{n-2}\}$. We show that $S_0$ is an edge metric generator of $G$. Let $e = uw$ and $f = yz$ be two non-adjacent edges in $G$. Then $|S_0\cap\{u, w, y, z\}|\geq 2$. Without loss of generality, we assume that $u\in S_0$. Since $d_G(e, u) = 0$, $d_G(f, u)\geq 1$, then $u$ distinguishes edges $e$ and $f$. Let $e' = tu_1$ and $f' = tu_2$ be two adjacent edges in $G$. If $u_1\in S_0$ (or $u_2\in S_0$), then $e'$ and $f'$ can be distinguished by $u_1$ (or $u_2$). Otherwise, we have $\{u_1, u_2\} = \{x, v_{n-1}\}$ and $t\in \{v_1, v_{n-2}\}$. First, we consider the case $t = v_1$. Since $n\geq 6$, then $S_0\setminus \{x, v_{n-1}, v_{n-2}, v_1, v_2\}\neq\emptyset$. It is easy to verify that $\{d_G(e', p), d_G(f', p)\} = \{1, 2\}$ for any vertex $p\in (S_0\setminus \{x, v_{n-1}, v_{n-2}, v_1, v_2\})$, which implies that $p$ distinguishes edges $e'$ and $f'$. Then we consider the case $t = v_{n-2}$. Since $n\geq 5$, then $S_0\setminus \{x, v_{n-1}, v_{n-2}, v_{n-3}, v_1\}\neq\emptyset$. Similarly, we have $\{d_G(e', q), d_G(f', q)\} = \{1, 2\}$ for any vertex $q\in (S_0\setminus \{x, v_{n-1}, v_{n-2}, v_{n-3}, v_1\})$ and then $q$ distinguishes edges $e'$ and $f'$. From the arbitrariness of $e, f$ and $e', f'$, we have that $S_0$ is an edge metric generator of $G$ and $edim(G)\leq n-2$, as desired.

In ^[4], Kelenc et al. compute the edge metric dimension for wheel graphs and fan graphs, which implies that the wheel graphs of order $n\geq 6$ and the fan graphs of order $n\geq 5$ have edge metric dimension $edim = n-2$. Actually, Proposition 2.1 generalized these results.

As is known to us, complete bipartite graph with $n$ vertices has edge metric dimension $n-2$. In the following theorem, we characterize all connected bipartite graphs with edge metric dimension $n-2$, which partially answers the open problem 1.

Theorem 2.1. Let $G$ be a connected bipartite graph of order $n\geq 3$. Then $edim(G) = n-2$ if and only if $G = K_{r, t}$, i.e., $G$ is a complete bipartite graph.

Proof. Let $G = G[U, V]$ be a connected bipartite graph with $U = \{u_1, u_2, \dots, u_r\}$ and $V = \{v_1, v_2, \dots, v_t\}$. If $G = K_{r, t}$ is a complete bipartite graph, then $edim(G) = n-2$ by Proposition 1.1.

Conversely, without loss of generality, we assume that $r\geq t$. If $t = 1$, then $G = K_{1, n-1} = S_{1, n-1}$ is a star graph and $edim(G) = n-2$. Now on we assume $t\geq 2$. Denote by $H$ a maximum complete bipartite induced subetaaph of $G$. Obviously, $|E(H)|\geq 2$ since $G$ is connected. Let $H_U = V(H)\cap U = \{u_{p+1}, u_{p+2}, \dots, u_r\}$ and $H_V = V(H)\cap V = \{v_{q+1}, v_{q+2}, \dots, v_t\}$ where $p, q$ are integers and $0\leq p\leq r-1$, $0\leq q\leq t-1$. Next it suffices to show that $G$ has an edge metric generator of size $n-3$ if $G$ is not complete bipartite, which implies that $H\neq G$ and $p+q\geq 1$.

If $\min\{|H_U|, |H_V|\}\geq 2$, then set $S = V(G)\setminus\{u_{p+1}, v_{q+1}, x\}$ such that there exists one vertex $h\in H_V\setminus\{v_{q+1}\}$, $hx\notin E(G)$ and $x\in V(G)\setminus(H_U\cup H_V)$. On one hand, $V(G)\setminus(H_U\cup H_V)\neq\emptyset$ since $p+q\geq 1$. Without loss of generality, set $x = u_1$. On the other hand, there exists some vertex $h\in H_V$ such that $hx\notin E(G)$ because $H$ is a maximum complete bipartite induced subetaaph of $G$. So an appropriate $v_{q+1}$ can be chosen after relabelling the vertices in $H_V$ if necessary. Next we will prove that $S$ is an edge metric generator of $G$. For any non-adjacent edges $e_1, e_2\in E(G)$, we have either $e_1$ or $e_2$ has one end vertex $y\in S$ and then $d_G(e_1, y) = 0, d_G(e_2, y)\geq 1$ or $d_G(e_2, y) = 0, d_G(e_1, y)\geq 1$. So $y$ distinguishes edges $e_1$ and $e_2$. For any adjacent edges $e_1 = zw, e_2 = zg$ of $G$, if $w\in S$ (or $g\in S$), then $e_1, e_2$ can be distinguished by $w$ (or $g$). Otherwise, we have $\{w, g\} = \{u_1, u_{p+1}\}$ and $z\in V\setminus \{h\}$. Since $hu_{p+1}\in E(G)$ and $hx = hu_1\notin E(G)$, then $\{d_G(e_1, h), d_G(e_2, h)\} = \{1, 2\}$ and $h$ distinguishes edges $e_1$ and $e_2$. Therefore, $S$ is an edge metric generator of $G$ and $edim(G)\leq n-3$.

If $\min\{|H_U|, |H_V|\} = 1$, then without loss of generality, we assume that $H_V = \{v_{t}\}$. Take $S = V(G)\setminus\{u_{p+1}, v_t, x\}$ such that there exists one vertex $h\in H_U\setminus\{u_{p+1}\}$, $hx\notin E(G)$ and $x\in V\setminus H_V$. On one hand, $V\setminus H_V\neq\emptyset$ since $r\geq t\geq 2$. On the other hand, there exists one vertex $h\in H_U$ such that $hx\notin E(G)$ because $H$ is a maximum complete bipartite induced subetaaph of $G$. So an appropriate $u_{p+1}$ can be chosen after relabelling the vertices in $H_U$ if necessary. With a similar analysis as in the case $\min\{|H_U|, |H_V|\}\geq 2$, we also have $S$ is an edge metric generator of $G$, which implies that $edim(G)\leq n-3$.

On one hand, Theorem 2.1 characterizes all connected bipartite graphs with edge metric dimension $n-2$. On the other hand, Proposition 2.1 implies that there are connected non-bipartite graphs with edge metric dimension $n-2$. So we will try to give a characterization of general connected graphs with edge metric dimension $n-2$.

Theorem 2.2. Let $G$ be a connected graph of order $n\geq 3$. If $H = H[U, V]$ is a complete bipartite spanning subetaaph of $G$ such that $|U| = s, |V| = t$, $s, t\geq 1$, then $edim(G) = n-2$ if and only if there is a vertex $u\in U$ such that for any $x\in N(u)\cap U$, $U\setminus(N(x)\cup N(u))\neq\emptyset$ if $N(u)\cap U\neq\emptyset$ and there is a vertex $v\in V$ such that for any $y\in N(v)\cap V$, $V\setminus(N(y)\cup N(v))\neq\emptyset$ if $N(v)\cap V\neq\emptyset$.

Proof. Let $H = H[U, V]$ be a complete bipartite spanning subetaaph of $G$ such that $|U| = s, |V| = t$, $s, t\geq 1$. Obviously, $s+t = n$.

First, we show that $edim(G)\geq n-2$. Let $S$ be an edge metric generator of $G$. Suppose that there are two vertices $u_1, u_2\in U\backslash S$. For each vertex $v\in V$, we have $d_G(u_1v, w) = d_G(u_2v, w) = 1$ for any vertex of $w\in V(G)\backslash\{u_1, u_2, v\}$, which contradicts to that $S$ is an edge metric generator of $G$. Therefore, $|U\backslash S|\leq 1$ and $|V\backslash S|\leq 1$. Thus $|S|\geq n-2$ and then $edim(G)\geq n-2$.

Next, we show that $edim(G)\leq n-2$ if and only if there is a vertex $u\in U$ such that for any $x\in N(u)\cap U$, $U\setminus(N(x)\cup N(u))\neq\emptyset$ if $N(u)\cap U\neq\emptyset$ and there is a vertex $v\in V$ such that for any $y\in N(v)\cap V$, $V\setminus(N(y)\cup N(v))\neq\emptyset$ if $N(v)\cap V\neq\emptyset$.

Without loss of generality, we assume that $s\geq t$. For the case $t = 1$, we have $s = n-t\geq 2$. Set $V = \{v_0\}$. If there is a vertex $u\in U$ such that for any $x\in N(u)\cap U$, $U\setminus(N(x)\cup N(u))\neq\emptyset$ if $N(u)\cap U\neq\emptyset$, then $S = V(G)\setminus\{u, v_0\}$ is an edge metric generator of $G$. On one hand, any pair of non-adjacent edges can be distinguished by some element of $S$ since $|S| = n-2$. On the other hand, we consider two adjacent edges $e_1 = zw_1$ and $e_2 = zw_2$. If $S\cap \{w_1, w_2\}\neq\emptyset$, then $e_1, e_2$ can be distinguished by any element of $S\cap \{w_1, w_2\}$. If $S\cap\{w_1, w_2\} = \emptyset$, then $\{w_1, w_2\} = \{u, v_0\}$ and $\{e_1, e_2\} = \{zu, zv_0\}$. Obviously, $z\in N(u)\cap U$ since $V = \{v_0\}$. For any element $p\in U\setminus(N(z)\cup N(u))$, we have $d_G(zv_0, p) = 1$, $d_G(zu, p) = 2$ and then each vertex in $U\setminus(N(z)\cup N(u))$ can distinguish edges $e_1, e_2$. Thus $S = V(G)\setminus\{u, v_0\}$ is an edge metric generator of $G$ and then $edim(G)\leq n-2$. Conversely, if $edim(G)\leq n-2$, then there is an edge metric generator $S$ of $G$ such that $|S| = n-2$ and $|U\setminus S| = 1$, $|V\setminus S| = 1$. Obviously, $V\setminus S = \{v_0\}$. Now we suppose $S = V(G)\setminus\{u, v_0\}$ and there is a vertex $x\in N(u)\cap U$ such that $U\setminus(N(x)\cup N(u)) = \emptyset$. Consider the edges $e_1 = xu, e_2 = xv_0$. For any element $z\in S$, we have $d_G(xu, z) = d_G(xv_0, z) = 1$ which contradicts to that $S$ is an edge metric generator of $G$. Therefore, there is a vertex $u\in U$ such that for any $x\in N(u)\cap U$, $U\setminus(N(x)\cup N(u))\neq\emptyset$ if $N(u)\cap U\neq\emptyset$.

Next we consider the case $s\geq t\geq 2$. If there is a vertex $u\in U$ such that for any $x\in N(u)\cap U$, $U\setminus(N(x)\cup N(u))\neq\emptyset$ if $N(u)\cap U\neq\emptyset$ and there is a vertex $v\in V$ such that for any $y\in N(v)\cap V$, $V\setminus(N(y)\cup N(v))\neq\emptyset$ if $N(v)\cap V\neq\emptyset$, then $S = V(G)\setminus\{u, v\}$ is an edge metric generator of $G$. Once $G[U] = \overline{K_s}$, then we choose any element of $U$ as $u$, so as for $v$. For any pair of non-adjacent edges, they can be distinguished by some element of $S$ since $|S| = n-2$. For two adjacent edges $e_1 = zw_1$ and $e_2 = zw_2$, if $S\cap \{w_1, w_2\}\neq\emptyset$, then $e_1, e_2$ can be distinguished by any element of $S\cap \{w_1, w_2\}$, otherwise $\{w_1, w_2\} = \{u, v\}$ and $\{e_1, e_2\} = \{zu, zv\}$. If $z\in U$, then for any element $p\in U\setminus(N(z)\cup N(u))$, we have $d_G(zv, p) = 1$, $d_G(zu, p) = 2$ and then each vertex in $U\setminus(N(z)\cup N(u))$ can distinguish edges $e_1, e_2$. For the case $z\in V$, edges $e_1, e_2$ can be distinguished by similar analysis. Thus $S = V(G)\setminus\{u, v\}$ is an edge metric generator of $G$ and then $edim(G)\leq n-2$. Conversely, if $edim(G)\leq n-2$, then there is an edge metric generator $S$ of $G$ such that $|S| = n-2$ and $|U\setminus S| = 1$, $|V\setminus S| = 1$. Now we suppose $S = V(G)\setminus\{u, v\}$. Without loss of generality, we assume that there is a vertex $x\in N(u)\cap U$ such that $U\setminus(N(x)\cup N(u)) = \emptyset$. Consider the edges $e_1 = xu, e_2 = xv$. Each element of $S$ has distance 1 to edges both $e_1$ and $e_2$, which comes to a contradiction. Therefore, there is a vertex $u\in U$ such that for any $x\in N(u)\cap U$, $U\setminus(N(x)\cup N(u))\neq\emptyset$ if $N(u)\cap U\neq\emptyset$. Similarly, there is a vertex $v\in V$ such that for any $y\in N(v)\cap V$, $V\setminus(N(y)\cup N(v))\neq\emptyset$ if $N(v)\cap V\neq\emptyset$ and thus we complete the proof.

One can check that Theorem 2.2 implies the results of Proposition 1.2 ^[4] and Proposition 1.3 ^[4]. Furthermore, the following corollary of Theorem 2.2 gives a sufficient and necessary condition to determine the value of $edim(G)$ for the graph $G$ with $n$ vertices and maximum degree $\Delta = n-1$.

Corollary 2.1. Let $G$ be a connected graph of order $n$. If there is a vertex $v\in V(G)$ of degree $n-1$, then $edim(G) = n-2$ if there is a vertex $u\in V(G)\setminus\{v\}$ such that for any $x\in N(u)\setminus\{v\}$, $(V(G)\setminus\{v\})\setminus(N(x)\cup N(u))\neq\emptyset$ if $N(u)\setminus\{v\}\neq\emptyset$ and otherwise $edim(G) = n-1$.

3.   The edge metric dimension and the clique number

Let $n > k\geq 1$ be positive integers. Denote by $\mathcal{G}$ the graph class consists of graphs $G_{n}^k$ which are obtained by identifying one vertex of $K_k$ and the center vertex of $S_{1, n-k}$, see Figure 1. Obviously, each element of $\mathcal{G}$ is a connected graph of order $n$.

Figure 1.  The graph $G_{n}^k$ of $\mathcal{G}$.

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Theorem 3.1. For any positive integer $p\geq 1$, there exists a connected graph $G$ such that $edim(G)-\omega(G) = p$.

Proof. Let $n > k\geq 1$ be positive integers such that $p = n-k-2$. Now we prove that $edim(G_{n}^k)-\omega(G_{n}^k) = p$ for any graph $G_{n}^k\in \mathcal{G}$. Obviously, it suffices to show that $edim(G_{n}^k) = n-2$.

Denote by $w$ as the vertex with degree $n-1$ in $G_n^k$ and $u$ as one of the pendent vertices in $G_n^k$. Then $edim(G_{n}^k)\geq n-2$ by Proposition 1.2. Next we will prove $edim(G_{n}^k)\leq n-2$. Set $S = V(G_{n}^k)\setminus\{w, u\}$. For any non-adjacent edges $e_1, e_2\in E(G_{n}^k)$, we have either $e_1$ or $e_2$ has one end vertex $y\in S$ and then $d_{G_{n}^k}(e_1, y) = 0, d_{G_{n}^k}(e_2, y)\geq 1$ or $d_{G_{n}^k}(e_2, y) = 0, d_{G_{n}^k}(e_1, y)\geq 1$. So $y$ distinguishes edges $e_1$ and $e_2$. For any adjacent edges $e_1 = zv, e_2 = zg$ of $G_{n}^k$, if $v\in S$ (or $g\in S$), then $e_1, e_2$ can be distinguished by $v$ (or $g$). Otherwise, we have $\{v, g\} = \{w, u\}$, which is impossible. Therefore, $S$ is an edge metric generator of $G_{n}^k$ and then $edim(G)\leq n-2$.

For the complete graph $K_n$, we have $edim(K_n) = n-1 = \omega(K_n)-1$, i.e., $\omega(K_n)-edim(K_n) = 1$. Naturally, what about the value of $\omega(G)-edim(G)$ for any connected graph $G\neq K_n$?

Problem 2. Let $G\neq K_n$ be a connected graph. Is it true that $\omega(G)-edim(G)\leq 1$, i.e., $edim(G)\geq \omega(G)-1$?

Let $k\geq 2$ be a positive integer. Denote by $\mathcal{H}$ the graph class consists of graphs $H_{2k}$ which are obtained by attaching one end of $K_2$ to each vertex of $K_{k}$. Obviously, each element of $\mathcal{H}$ is a connected graph of order $2k$ and clique number $k$.

We claim that $edim(H_{2k}) = k-1 = \omega(H_{2k})-1$, where $H_{2k}$ is depicted in Figure 2. It suffices to show that $S = \{v_1, v_2, \dots, v_{k-1}\}$ is an edge metric generator of $H_{2k}$. For any non-adjacent edges $e_1 = w_1w_2, e_2 = w_3w_4$ of $H_{2k}$, if $S\cap \{w_1, w_2, w_3, w_4\}\neq\emptyset$, then any element of $S\cap \{w_1, w_2, w_3, w_4\}$ can distinguish edges $e_1$ and $e_2$. Otherwise, we have $e_1$ or $e_2$ or both of them are adjacent to a pendent edge $u_iv_i$ such that $v_i\in S$ and then $d(e_1, v_i) = 1, d(e_2, v_i) = 2$ or $d(e_1, v_i) = 2, d(e_2, v_i) = 1$. So $v_i$ distinguishes edges $e_1$ and $e_2$. For any adjacent edges $e_1 = w_5w_6, e_2 = w_5w_7$ of $H_{2k}$, if $w_6\in S$ (or $w_7\in S$), then $e_1, e_2$ can be distinguished by $w_6$ (or $w_7$). Otherwise, we have $\{w_6, w_7\}\subseteq\{u_1, u_2, \dots, u_k, v_k\}$. Hence, there exists $v_j\in S$ such that $d(e_1, v_j) = 1, d(e_2, v_j) = 2$ or $d(e_1, v_j) = 2, d(e_2, v_j) = 1$. Therefore, $S$ is an edge metric generator of $H_{2k}$ and then $edim(H_{2k}) = k-1 = \omega(H_{2k})-1$. This implies that once the answer of Problem 2 is true, then the lower bound $edim(G)\geq \omega(G)-1$ is sharp.

Figure 2.  The graph $H_{2k}$ of $\mathcal{H}$.

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Acknowledgments

The authors would like to thank the anonymous referees for many helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 11626148 and 11701342) and the Natural Science Foundation of Shandong Province (No. ZR2016AQ01 and ZR2019MA032).

Conflict of interest

The authors declare no conflict of interest.