Research article

On the edge metric dimension of graphs

  • Received: 17 March 2020 Accepted: 10 May 2020 Published: 18 May 2020
  • MSC : 05C40

  • Let G=(V,E) be a connected graph of order n. SV is an edge metric generator of G if any pair of edges in E can be distinguished by some element of S. The edge metric dimension edim(G) of a graph G is the least size of an edge metric generator of G. In this paper, we give the characterization of all connected bipartite graphs with edim=n2, which partially answers an open problem of Zubrilina (2018). Furthermore, we also give a sufficient and necessary condition for edim(G)=n2, where G is a graph with maximum degree n1. In addition, the relationship between the edge metric dimension and the clique number of a graph G is investigated by construction.

    Citation: Meiqin Wei, Jun Yue, Xiaoyu zhu. On the edge metric dimension of graphs[J]. AIMS Mathematics, 2020, 5(5): 4459-4465. doi: 10.3934/math.2020286

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  • Let G=(V,E) be a connected graph of order n. SV is an edge metric generator of G if any pair of edges in E can be distinguished by some element of S. The edge metric dimension edim(G) of a graph G is the least size of an edge metric generator of G. In this paper, we give the characterization of all connected bipartite graphs with edim=n2, which partially answers an open problem of Zubrilina (2018). Furthermore, we also give a sufficient and necessary condition for edim(G)=n2, where G is a graph with maximum degree n1. In addition, the relationship between the edge metric dimension and the clique number of a graph G is investigated by construction.


    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 vV, the open neighborhood of v is N(v)={uV:uvE} and the closed neighborhood of v is N[v]=N(v){v}. The degree of a vertex vV is defined as dG(v)=|N(v)|. The minimum and maximum degree of a graph G are denoted by δ=δ(G) and Δ=Δ(G), respectively. The distance between the vertex wV and the subetaaph HG is defined as dG(w,H)=min{dG(w,v)|vV(H)}. Particularly, when H is an edge e=uvE, then dG(w,e)=min{dG(w,u),dG(w,v)}. A vertex wV distinguishes two edges e1,e2E if dG(e1,w)dG(e2,w). We say SV(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, 1edim(G)n1 are natural bounds on the edge metric dimension. In [4], Kelenc et al. proved that edim(G)=n2 when G is a wheel graph of order n6 or a fan graph of order n5. And further, they also gave the lower bound of edge metric dimension edim(G)n2 for any connected graph G with Δ=n1, 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=n1 and edim=n2. Recently, Zubrilina [6] classified the graphs on n vertices for which edim(G)=n1. They also proposed an open problem as follows.

    Problem 1. For which graphs G of order n is edim(G)=n2?

    In this paper, we consider the first step for this problem on connected bipartite graphs. And then we classify the class of graphs of Δ=n1 with edim=n2.

    In [8], Zhu et al. constructed connected graphs G of order n such that edim(G)=n2 and edim(Ge)=n1, where eE(G). This implies that for any subetaaph HG, 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 HG. And further, we also consider the relation between edim(G) and clique number ω(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 Kr,t different from K1,1, edim(Kr,t)=dim(Kr,t)=r+t2.

    Proposition 1.2. [4] Let G be a connected graph of order n. If there is a vertex vV(G) of degree n1, then either edim(G)=n1 or edim(G)=n2.

    Proposition 1.3. [4] Let G be a connected graph of order n. If there are two vertices u,vV(G) of degree n1, then edim(G)=n1.

    Given two graphs G and H, the join, denoted by GH, 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 W1,n is isomorphic to CnK1, the fan graph F1,n is isomorphic to PnK1 and the star graph S1,n is isomorphic to ¯KnK1. The vertex of degree n in the star graph S1,n is denoted as its center vertex.

    Proposition 2.1. Let W1,n1=Cn1K1 be a wheel graph and G be a graph obtained from W1,n1 by deleting any edges on the cycle Cn1, where n6. Then edim(G)=n2.

    Proof. Suppose n6 and V(W1,n1)={x,v1,v2,,vn2,vn1}, where the vertex x has degree n1 and the vertices v1,v2,,vn2,vn1 induce the cycle Cn1.

    Let S be an edge metric generator of G. If there exist vi,vj(V(Cn1)S), then the edges xvi and xvj can not be distinguished by any vertex wV(G){vi,vj} because dG(xvi,w)=dG(xvj,w)=1 or dG(xvi,w)=dG(xvj,w)=0. So |V(Cn1)S|1 and then |S|n2. From the arbitrariness of S, we have edim(G)n2.

    Next, it suffices to show that edim(G)n2. Set S0={v1,v2,,vn2}. We show that S0 is an edge metric generator of G. Let e=uw and f=yz be two non-adjacent edges in G. Then |S0{u,w,y,z}|2. Without loss of generality, we assume that uS0. Since dG(e,u)=0, dG(f,u)1, then u distinguishes edges e and f. Let e=tu1 and f=tu2 be two adjacent edges in G. If u1S0 (or u2S0), then e and f can be distinguished by u1 (or u2). Otherwise, we have {u1,u2}={x,vn1} and t{v1,vn2}. First, we consider the case t=v1. Since n6, then S0{x,vn1,vn2,v1,v2}. It is easy to verify that {dG(e,p),dG(f,p)}={1,2} for any vertex p(S0{x,vn1,vn2,v1,v2}), which implies that p distinguishes edges e and f. Then we consider the case t=vn2. Since n5, then S0{x,vn1,vn2,vn3,v1}. Similarly, we have {dG(e,q),dG(f,q)}={1,2} for any vertex q(S0{x,vn1,vn2,vn3,v1}) and then q distinguishes edges e and f. From the arbitrariness of e,f and e,f, we have that S0 is an edge metric generator of G and edim(G)n2, 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 n6 and the fan graphs of order n5 have edge metric dimension edim=n2. Actually, Proposition 2.1 generalized these results.

    As is known to us, complete bipartite graph with n vertices has edge metric dimension n2. In the following theorem, we characterize all connected bipartite graphs with edge metric dimension n2, which partially answers the open problem 1.

    Theorem 2.1. Let G be a connected bipartite graph of order n3. Then edim(G)=n2 if and only if G=Kr,t, i.e., G is a complete bipartite graph.

    Proof. Let G=G[U,V] be a connected bipartite graph with U={u1,u2,,ur} and V={v1,v2,,vt}. If G=Kr,t is a complete bipartite graph, then edim(G)=n2 by Proposition 1.1.

    Conversely, without loss of generality, we assume that rt. If t=1, then G=K1,n1=S1,n1 is a star graph and edim(G)=n2. Now on we assume t2. Denote by H a maximum complete bipartite induced subetaaph of G. Obviously, |E(H)|2 since G is connected. Let HU=V(H)U={up+1,up+2,,ur} and HV=V(H)V={vq+1,vq+2,,vt} where p,q are integers and 0pr1, 0qt1. Next it suffices to show that G has an edge metric generator of size n3 if G is not complete bipartite, which implies that HG and p+q1.

    If min{|HU|,|HV|}2, then set S=V(G){up+1,vq+1,x} such that there exists one vertex hHV{vq+1}, hxE(G) and xV(G)(HUHV). On one hand, V(G)(HUHV) since p+q1. Without loss of generality, set x=u1. On the other hand, there exists some vertex hHV such that hxE(G) because H is a maximum complete bipartite induced subetaaph of G. So an appropriate vq+1 can be chosen after relabelling the vertices in HV if necessary. Next we will prove that S is an edge metric generator of G. For any non-adjacent edges e1,e2E(G), we have either e1 or e2 has one end vertex yS and then dG(e1,y)=0,dG(e2,y)1 or dG(e2,y)=0,dG(e1,y)1. So y distinguishes edges e1 and e2. For any adjacent edges e1=zw,e2=zg of G, if wS (or gS), then e1,e2 can be distinguished by w (or g). Otherwise, we have {w,g}={u1,up+1} and zV{h}. Since hup+1E(G) and hx=hu1E(G), then {dG(e1,h),dG(e2,h)}={1,2} and h distinguishes edges e1 and e2. Therefore, S is an edge metric generator of G and edim(G)n3.

    If min{|HU|,|HV|}=1, then without loss of generality, we assume that HV={vt}. Take S=V(G){up+1,vt,x} such that there exists one vertex hHU{up+1}, hxE(G) and xVHV. On one hand, VHV since rt2. On the other hand, there exists one vertex hHU such that hxE(G) because H is a maximum complete bipartite induced subetaaph of G. So an appropriate up+1 can be chosen after relabelling the vertices in HU if necessary. With a similar analysis as in the case min{|HU|,|HV|}2, we also have S is an edge metric generator of G, which implies that edim(G)n3.

    On one hand, Theorem 2.1 characterizes all connected bipartite graphs with edge metric dimension n2. On the other hand, Proposition 2.1 implies that there are connected non-bipartite graphs with edge metric dimension n2. So we will try to give a characterization of general connected graphs with edge metric dimension n2.

    Theorem 2.2. Let G be a connected graph of order n3. If H=H[U,V] is a complete bipartite spanning subetaaph of G such that |U|=s,|V|=t, s,t1, then edim(G)=n2 if and only if there is a vertex uU such that for any xN(u)U, U(N(x)N(u)) if N(u)U and there is a vertex vV such that for any yN(v)V, V(N(y)N(v)) if N(v)V.

    Proof. Let H=H[U,V] be a complete bipartite spanning subetaaph of G such that |U|=s,|V|=t, s,t1. Obviously, s+t=n.

    First, we show that edim(G)n2. Let S be an edge metric generator of G. Suppose that there are two vertices u1,u2US. For each vertex vV, we have dG(u1v,w)=dG(u2v,w)=1 for any vertex of wV(G){u1,u2,v}, which contradicts to that S is an edge metric generator of G. Therefore, |US|1 and |VS|1. Thus |S|n2 and then edim(G)n2.

    Next, we show that edim(G)n2 if and only if there is a vertex uU such that for any xN(u)U, U(N(x)N(u)) if N(u)U and there is a vertex vV such that for any yN(v)V, V(N(y)N(v)) if N(v)V.

    Without loss of generality, we assume that st. For the case t=1, we have s=nt2. Set V={v0}. If there is a vertex uU such that for any xN(u)U, U(N(x)N(u)) if N(u)U, then S=V(G){u,v0} 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|=n2. On the other hand, we consider two adjacent edges e1=zw1 and e2=zw2. If S{w1,w2}, then e1,e2 can be distinguished by any element of S{w1,w2}. If S{w1,w2}=, then {w1,w2}={u,v0} and {e1,e2}={zu,zv0}. Obviously, zN(u)U since V={v0}. For any element pU(N(z)N(u)), we have dG(zv0,p)=1, dG(zu,p)=2 and then each vertex in U(N(z)N(u)) can distinguish edges e1,e2. Thus S=V(G){u,v0} is an edge metric generator of G and then edim(G)n2. Conversely, if edim(G)n2, then there is an edge metric generator S of G such that |S|=n2 and |US|=1, |VS|=1. Obviously, VS={v0}. Now we suppose S=V(G){u,v0} and there is a vertex xN(u)U such that U(N(x)N(u))=. Consider the edges e1=xu,e2=xv0. For any element zS, we have dG(xu,z)=dG(xv0,z)=1 which contradicts to that S is an edge metric generator of G. Therefore, there is a vertex uU such that for any xN(u)U, U(N(x)N(u)) if N(u)U.

    Next we consider the case st2. If there is a vertex uU such that for any xN(u)U, U(N(x)N(u)) if N(u)U and there is a vertex vV such that for any yN(v)V, V(N(y)N(v)) if N(v)V, then S=V(G){u,v} is an edge metric generator of G. Once G[U]=¯Ks, 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|=n2. For two adjacent edges e1=zw1 and e2=zw2, if S{w1,w2}, then e1,e2 can be distinguished by any element of S{w1,w2}, otherwise {w1,w2}={u,v} and {e1,e2}={zu,zv}. If zU, then for any element pU(N(z)N(u)), we have dG(zv,p)=1, dG(zu,p)=2 and then each vertex in U(N(z)N(u)) can distinguish edges e1,e2. For the case zV, edges e1,e2 can be distinguished by similar analysis. Thus S=V(G){u,v} is an edge metric generator of G and then edim(G)n2. Conversely, if edim(G)n2, then there is an edge metric generator S of G such that |S|=n2 and |US|=1, |VS|=1. Now we suppose S=V(G){u,v}. Without loss of generality, we assume that there is a vertex xN(u)U such that U(N(x)N(u))=. Consider the edges e1=xu,e2=xv. Each element of S has distance 1 to edges both e1 and e2, which comes to a contradiction. Therefore, there is a vertex uU such that for any xN(u)U, U(N(x)N(u)) if N(u)U. Similarly, there is a vertex vV such that for any yN(v)V, V(N(y)N(v)) if N(v)V 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 Δ=n1.

    Corollary 2.1. Let G be a connected graph of order n. If there is a vertex vV(G) of degree n1, then edim(G)=n2 if there is a vertex uV(G){v} such that for any xN(u){v}, (V(G){v})(N(x)N(u)) if N(u){v} and otherwise edim(G)=n1.

    Let n>k1 be positive integers. Denote by G the graph class consists of graphs Gkn which are obtained by identifying one vertex of Kk and the center vertex of S1,nk, see Figure 1. Obviously, each element of G is a connected graph of order n.

    Figure 1.  The graph Gkn of G.

    Theorem 3.1. For any positive integer p1, there exists a connected graph G such that edim(G)ω(G)=p.

    Proof. Let n>k1 be positive integers such that p=nk2. Now we prove that edim(Gkn)ω(Gkn)=p for any graph GknG. Obviously, it suffices to show that edim(Gkn)=n2.

    Denote by w as the vertex with degree n1 in Gkn and u as one of the pendent vertices in Gkn. Then edim(Gkn)n2 by Proposition 1.2. Next we will prove edim(Gkn)n2. Set S=V(Gkn){w,u}. For any non-adjacent edges e1,e2E(Gkn), we have either e1 or e2 has one end vertex yS and then dGkn(e1,y)=0,dGkn(e2,y)1 or dGkn(e2,y)=0,dGkn(e1,y)1. So y distinguishes edges e1 and e2. For any adjacent edges e1=zv,e2=zg of Gkn, if vS (or gS), then e1,e2 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 Gkn and then edim(G)n2.

    For the complete graph Kn, we have edim(Kn)=n1=ω(Kn)1, i.e., ω(Kn)edim(Kn)=1. Naturally, what about the value of ω(G)edim(G) for any connected graph GKn?

    Problem 2. Let GKn be a connected graph. Is it true that ω(G)edim(G)1, i.e., edim(G)ω(G)1?

    Let k2 be a positive integer. Denote by H the graph class consists of graphs H2k which are obtained by attaching one end of K2 to each vertex of Kk. Obviously, each element of H is a connected graph of order 2k and clique number k.

    We claim that edim(H2k)=k1=ω(H2k)1, where H2k is depicted in Figure 2. It suffices to show that S={v1,v2,,vk1} is an edge metric generator of H2k. For any non-adjacent edges e1=w1w2,e2=w3w4 of H2k, if S{w1,w2,w3,w4}, then any element of S{w1,w2,w3,w4} can distinguish edges e1 and e2. Otherwise, we have e1 or e2 or both of them are adjacent to a pendent edge uivi such that viS and then d(e1,vi)=1,d(e2,vi)=2 or d(e1,vi)=2,d(e2,vi)=1. So vi distinguishes edges e1 and e2. For any adjacent edges e1=w5w6,e2=w5w7 of H2k, if w6S (or w7S), then e1,e2 can be distinguished by w6 (or w7). Otherwise, we have {w6,w7}{u1,u2,,uk,vk}. Hence, there exists vjS such that d(e1,vj)=1,d(e2,vj)=2 or d(e1,vj)=2,d(e2,vj)=1. Therefore, S is an edge metric generator of H2k and then edim(H2k)=k1=ω(H2k)1. This implies that once the answer of Problem 2 is true, then the lower bound edim(G)ω(G)1 is sharp.

    Figure 2.  The graph H2k of H.

    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).

    The authors declare no conflict of interest.



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