SciresolSciresolhttps://indjst.org/author-guidelinesIndian Journal of Science and Technology0974-564510.17485/IJST/v17i13.3114Research articleLict k-Domination in GraphsGadeSuahas P1VyavahareDayanand Ksuhaspanduranggade@gmail.com2Assistant Professor, Department of Mathematics, Sangameshwar College (Autonomous)Solapur, MaharashtraIndiaResearch Scholar, School of Computational Sciences, PAH Solapur UniversitySolapur, MaharashtraIndia22320241713131591220231320242024
<bold id="s-947a3065acc5">Abstract</bold>

Objectives: In this study, we find lict k-domination numberγknGof various types of graphs. Methods: Let G be any graph, a set D⊆VnG is said to be an k-dominating set of lict graph n(G) if every vertex v∈VnG-D is dominated by at least k vertices in D, that is Nv∩D≥k. The Lict k-domination number γkn(G)is the minimum cardinality of k-dominating set of n(G). Findings: This study is centered on the lict k-domination number γkn(G) of the graph G and developed its relationship with other different domination parameters. Novelty: This study introduces the concept of “Lict k-Domination in Graphs”. It obtains many bounds on γkn(G) in terms of vertices, edges, and other different parameters of G.

KeywordsDomination numberk-domination numberLict graphLict k-domination numberTotal domination numberIndependent domination numberCouncil of Scientific and Industrial Research Human Resource Development Group (CSIR-HRDG), grant no. 09/0990(11223)/2021-EMR-IIntroduction

This study considers simple, finite, non-trivial, undirected and connected graph denoted as GV,E=(p,q), where V represents the vertex set and E denotes the edge set of the graph. The order and size of graph G is the VG=p and EG=q. For our notation and corresponding definitions, we follow the F. Harary 1. A lict graph 2nG associated with graph G is constructed with a vertex set that is the union of the sets of edges and cut vertices of graph G. In this context, two vertices in n(G) are adjacent if and only if their corresponding edges in graph G are adjacent, or the corresponding elements of G are incident (Figure 1).

We begin by recalling some definitions from domination theory.

A set D⊆V(G) is a dominating set if for each vertex u∈D⊆V(G),ND∩D≠ϕ.The domination number γ(G) is the minimum cardinality of a dominating set of G. If every vertex inV(G) is connected to at least one neighbor in D, then the subset D⊆V(G)is defined as a total dominating set. The total domination number γt(G) is represents the cardinality of a minimum total dominating set^{ 3. }A subset D⊆V(G)is an independent dominating set if the induced subgraph <D> contains no edges. The independence domination number γi(G) is corresponds to the cardinality of a minimum independence dominating set (see 4).

<bold id="s-5830c72ebb90">Related work</bold>

Many authors studied different domination in lict graph of graph. In 5 author introduce the new concept weak domination in lict graphs and determine upper and lower bounds for γwn(G) in terms of parameter graph G. In the context of 6 the focus is on set domination in lict graphs and develop the properties of the set-domination. Also, relate γs(G) with some domination parameter. In7 studied Split Domination in Lict Subdivision Graph of a Graph and many bounds were obtained in terms of the various element of G. Additionally, they investigated relation between other different domination parameters of graph. Also, in Lict Subdivision Connected Domination in Graphs8 work onγnsc(G) to relate this with the components of G.

The Concept of k-domination and its generalization as total k-domination number γkt(G) used in 9 strong product graph to obtain several bounds for the γkt(G) of the strong product of two graphs. As in 2020, Ekinci G.B., Bujtas C. characterized bipartite graphs satisfying the equality for k≥3 and presented a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily if k=3^{10}. In 2022 ^{11}, author studied relationships between the k-tuple domination number and parameters of graph and different dominations. Also, ^{12} studied the total k-domination number of Cartesian product of two complete graphs and obtained some lower and upper bounds for the total k-domination number of Cartesian product of two complete graphs.

This work uses the concept of k-domination within the lict graph of graph G. The lict k-domination number denoted as γkn(G), represents the minimum cardinality among all k-dominating set of lict graph n(G). Our investigation yields numerous bounds on γkn(G) in terms various parameters such as vertices, edges, and other distinct parameters of G. Additionally, this study establishes relationships between γkn(G) other different domination parameters of G.

<bold id="s-cada6d59f1a2">Methodology</bold>

Theorem 2.1.^{ }13, 14^{ }A subset D⊆V(G)is k-dominating set if every vertex of D⊆V(G) is adjacent to at least kvertices of D. The k-domination number γk(G) represents the minimum cardinality of k-dominating set of G.

Theorem 2.2.^{ }15 For any Tree T, γse(n(T))≤c, where c is the number of cut-vertices of T.

Theorem 2.3.^{ }16 For any connected graph G, γcG≥γ(G).

Theorem 2.4.^{ }16 For any connected graph G with p≥2vertices,γct(G)≥γt(G).

Theorem 2.5.^{ }15 For any connected graph G,γse(n(G))≥γ'(G).

Theorem 2.6.^{ }17 For any graph G,γt(n(G))≥γ'(G).

<bold id="s-e7ddad5d1eb4">Results and Discussion</bold>

Theorem 3.1: If G=(p,q)be any graph, thenγkn(G)≥q+c△'nG+1.

Proof. Let V(nG) be the set u1,u2,…,uq+c, representing the vertex set of the lict graph of graph G. There exists a minimal set A1=u1,u2,…,uq+c⊆V(n(G)), such that every vertex in Vn(G)-A1 is adjacent to at least kvertices of A1. This establishes A1 as the minimal k-dominating set of n(G). Clearly, A1=γkn(G).

It gives that each vertex in Vn(G)-A1 contributes at least one to the sum of degree of vertices of A1. This implies that the size of set (Vn(G)-A1 is less than or equal to the sum of the degrees of vertices in A1, given by:

VnG-A1≤∑ui∈A1deg(ui)

In the lict graphn(G), there exists at least one vertex uj∈V(n(G)) such that degui=△'nG.

We can conclude that,

q+c-γknG=VnG-A1≤∑ui∈A1deg(ui)

≤γknG*△'nG.

Therefore, q+c-γknG≤γknG*△'nG.

Thus, γkn(G)≥q+c△'nG+1.

Theorem 3.2: If G=(p,q) be any graph with 2≤k≤△(G), then γknG≥γG+k-2.

Proof. Let V(G)=u1,u2,…,up represent the vertex set a graph G. Then, there is subset D of V(G) that covers all the vertices of G. This shows that D is dominating set of G.Consider VnG=w1,w2,…,wq+c as the vertex set of lict graph of graph G, and let S⊆V(n(G)) be a minimum k-dominating set of n(G).

Now, take w∈VnG-S, and let w1,w2,…,wk be distinct vertices in Swhich dominate w. Since S is a k-dominating set, each vertex in VnG-S is dominated by at least one vertex in S-w2,w3,…,wk. Therefore, w dominates each vertex in w2,w3,…,wk.

Which implies,

D≤S-w2,w3,…,wk∪w

γG≤γknG-k-1+1

=γknG-k+2

Thus, γknG≥γG+k-2.

Theorem 3.3: For any graph G=(p,q) and a positive integer k≥2, then γknG≥c. Where c is number of cut vertices of G.

Proof.Case-I: If the graph has no cut vertex, then result holds trivially.

Case-II: If the graph has cut vertices.

Assume A1=m1,m2,…,mc be subset of a vertex set of n(G) contains cut vertices of G. Let VnG=w1,w2,…,wq,m1,m2,…,mcbe a vertex set of n(G), where w1,w2,…,wq corresponds to edges incident on vertices of graph G.

Consider a subset D1 of VnG,which contains vertices of n(G) which corresponding to edges incident with cut vertices of graph G. Then, D1 forms a minimal k-dominating set of lict graph of G. Otherwise, we can add cut vertices in D1 to get a k-dominating set of VnG. A cardinality of such a set is greater or equal to the number of cut vertices of G.

Therefore, γknG≥c.

Corollary 3.4: For any tree T and a positive integer k≥2 then γknT≥γsen(T).

Proof. From Theorem 2.2, we have

γsen(T)≤c

From Theorem 3.3, we get

γknG≥c

From , , we Conclude that, this is required result.

Theorem 3.5: If G=p,qis any graph and k is a positive integer with k≥2, then γknG≥γiG.

Proof. Let A=u1,u2,…,usbe a dominating set of G. If the induces subgraph <A> contains vertices of degree zero, then A itself is an independent dominating set. Otherwise, let S=A1∪A2, where A1⊆A and A2⊆VG-A forming a minimal independent dominating set of G such that S=γiG.

Now, consider D=e1,e2,…,es,m1,m2,…,mt,a subset of vertex set of VnG, where ei's are edges incident on vertices of set S and mi's are the cut vertices of G. If every vertex of VnG-Dis adjacent to at least kvertices of D then D itself is a k-dominating set of n(G). Otherwise, let S1=D1∪D2, where D1⊆D and D2⊆Vn(G)-D, forms minimal k-dominating set of nG, and it gives S≤S1.

Thus, γknG≥γiG.

Theorem 3.6: If G=p,qis any graph and k is a positive integer with k≤△(G), then γknG+γn(G)≥q+c.

Proof. For k≤△(G), let S=w1,w2,…,wr⊆VnG be minimal k-dominating set of n(G). Assume that there exists vertex w∈Sthat is not adjacent to any vertex in Vn(G)-S. This implies that w is adjacent to at least k vertices of set S. This Shows that the set S-w is minimal k-dominating set of n(G). Which contradicts the set S is minimal k-dominating set. Therefore, every vertex in S is adjacent to at least one vertex of Vn(G)-S. This implies Vn(G)-S is dominating set of n(G). Clearly, there exists set D⊆Vn(G)-S] is minimal dominating set of n(G) such that D+S≤q+c.

Thus, γknG+γn(G)≥q+c.

Theorem 3.7: If G=p,qis any graph. Then, γ'(G)≤γn(G)≤γknG.

Proof. Case-I: If the graph has no cut vertex, the result holds obviously.

Case-II: If the graph has cut vertices.

Let E(G)=e1,e2,…,eq be edge set of G, and C(G)=m1,m2,…,mc be set of cut vertices of G. The vertex set of lict graph is VnG=E(G)∪C(G). Let E1=e1,e2,…,es⊆EGbe minimal edge dominating set of G such that E=γ'(G). Since E(G)⊆VnG, if set E1 forms a dominating set of n(G) then result holds. Otherwise, let A=VnG-E1, then A1∪E2, where A1⊆A and E2⊆E1 covers all vertices of n(G), and which is minimal dominating set of n(G).

Clearly, |E1|≤|A1∪E2|.

Therefore, γ'(G)≤γn(G). Also, we get required result.

Theorem 3.8: If G=p,qis any graph. Then, γknG≤q+α0(G).

Proof. Case-I: If the graph has no cut vertex, the result holds obviously.

Case-II: If the graph has cut vertices.

As VnG=w1,w2,…,wq+ccontains q edges and c cut vertices. Let VnG=w1,w2,…,wq+c Subset V(nG) corresponding to edges of graph G. A set A=u1,u2,…,us be the vertex cover of G, which also contain a cut vertices of graph G. Consider E1⊆E and A1⊆A, then A1∪E1 forms minimal a k-dominating set of n(G) such that A1∪E1≤E+A.

We get, γknG≤q+α0(G).

Theorem 3.9: Let G=p,q be any graph. Then, γknG≤p+α1(G).

Proof. Let A=u1,u2,…,up be vertex set of graph G. A set B=e1,e2,…,es is edge cover of graph G such that B=α1(G).

As VnG=w1,w2,…,wq+c is the vertex set of n(G), which is a collection of edges and cut vertices of G. We take a set D1=w1,w2,…,wqbe the edges incident on vertices of G, and C1=wq+1,wq+2,…,wq+i are the cut vertices adjacent to element of edge cover of G. Consider D2⊆D1 and C2⊆C1, then D2∪C2 forms minimal a k-dominating set of nG.

Clearly, D2∪C2≤A+B.

Thus, γknG≤p+α1(G).

Theorem 3.10: Let G=p,q be any graph. Then, γknG<q+diam(G).

Proof. Case-I: If the graph has no cut vertex, the vertex set of lict graph contains vertices, which corresponds to edges of G, such that V(n(G))=q. This follows the result.

Case-II: If the graph has cut vertices.

As vertex set of n(G) is VnG=E(G)∪C(G), where E(G) is the edge set of G and C(G) is set of cut vertices of G. Let E=w1,w2,…,wq subset V(n(G)) corresponding to the edges of graph G. A set D=wq+1,wq+2,…,wq+c be cut vertices of G, which are lie on the longest path between two vertices u1andu2of G such that distu1,u2=diam(G). Consider E1⊆E and D1⊂D, then E1∪D1 forms minimal a k-dominating set of nG.

It follows that, E1∪D1<E+distu1,u2.

We get, γknG<q+diam(G).

Theorem 3.11: If G=p,qis any graph. Then, γknG<q+β0(G).

Proof. Case-I: If the graph has no cut vertex, the result holds.

Case-II: If the graph has cut vertices.

A vertex set of n(G) is a collection those vertices which corresponds to edges and cut vertices of G. Let E=w1,w2,…,wqsubset V(n(G)) corresponding to edges of graph G, and set B=u1,u2,…,us be the maximum independent set of G such thatB=β0(G). Consider set C=wq+1,wq+2,…,wq+c contain cut vertices of G which are either member of B or adjacent to elements of B such that C<B.

Let E1⊆E and C1⊂C, then E1∪C1 forms minimal a k-dominating set of nG.

Clearly, E1∪C1<E+B.

Thus, γknG<q+β0(G).

Theorem 3.12: If G=p,qis any graph. Then, γknG≤q+γt(G).

Proof. Case-I: If the graph has no cut vertex, then result holds.

Case-II: If the graph has cut vertices.

Let D=u1,u2,…,us be a dominating set of G and D1=VG-D be a set such that H⊆D1with a minimum set of vertices. If <D> does not contain an isolated vertex, then D itself total dominating set of G. Otherwise, some ui∈H such that ∀uj∈D, (ui,uj)∈E(G)

and <D1>=<D∪ui> has no isolated vertex. Then D1=D∪ui is a total dominating set of G such that D1=γt(G).

Let E=w1,w2,…,wqsubset V(n(G)) corresponding to edges of graph G. A set D1=u1,u2,…,ut be a minimal total dominating set, and the cut vertices are also member of such set. Let E1⊆E and D2⊆D1, then E1∪D2 forms minimal a k-dominating set of nG. It gives, E1∪D2≤E+D1.

Hence, the result.

Theorem 3.13: If G=p,qis any graph and k is a positive integer with k≥2, then γknG≥p△'(G)

Proof. Let V(G)=u1,u2,…,up be vertex set of G, and E(G)=e1,e2,…,eq are edges incident on vertices of G. Which are members of vertex set of n(G), and set E(G)=e1,e2,…,eq cut vertices of G.

For k≥2,VnG=E(G)∪C(G) be a vertex set of lict graph of G. Then, there exists a set B such that every vertex of VnG-B adjacent to at least k vertices of B. This implies that B forms minimal k-dominating set of n(G). for every graph, there exists an edge e∈E(G) of maximum degree, i.e., dege=△'(G) Such that, △'G*|B|≥|V(G)|.

∴△'G*γknG≥p

Thus, γknG≥p△'G.

Theorem 3.14:If G=p,qis any graph, then γknG+γtG≤q+p.

Proof.Case-I: If the graph has no cut vertex, then γknG≤q and γtG≤p. We get required result.

Case-II: If the graph has cut vertices.

Let V(G)=u1,u2,…,up be vertex set of G then there exists a set A=u1,u2,…,us ⊆ V(G) such that <A> has a no isolated vertex. This shows A is total dominating set of G and A=γtG.

Let set C=m1,m2,…,mcbe the cut vertices of G and E=e1,e2,…,eq be edge set of G. Which are the members of vertex set of nG. Consider E1⊆E and C1⊆C, then E1∪C1 forms minimal a k-dominating set of nGsuch that E1∪C1+A≤E(G)+V(G).

Thus, γknG+γtG≤q+p.

Theorem 3.15: If G=p,qis any graph, then γknG+γG≥γtG.

Proof. Case-I: If γG=γtG, then the result holds clearly.

Case-II: If γG≠γtG.

Let D=u1,u2,…,up be a dominating set of G and <D> contain isolated vertex. Then, there exists ui∈VG-D such that ∀uj∈D, (ui,uj)∈E(G) and <D1>=<D∪ui>has no isolated vertex. Then, D1=D∪ui is total dominating set of G. Which contains cut vertices of Gand V(n(G)) be vertex set of n(G) contains vertices which correspond to edges and cut vertices of G. So, there is minimal set A⊆V(n(G)) which is k-dominating set of n(G) such that, A+D≥D1.

We get, γknG+γG≥γtG.

Corollary 3.16: If G=p,qis any graph, then γknG+γcG.≥γtG.

Proof. Using Theorem 2.2 and Theorem 3.16, we get required result.

Theorem 3.17: If G=p,qis any graph, then γknG+γiG≤q+p.

Proof. Case-I: If the graph has no cut vertex, then γknG≤q and γiG≤p. We get required result.

Case-II: If the graph has cut vertices.

Let A=u1,u2,…,us be dominating set of G. If induced subgraph <A> contains vertices of degree zero, then set A itself minimal independent dominating set of G. Otherwise, S=A1∪A2, where A1⊆Aand A2⊆VG-A forms minimal independent dominating set of G such that S=γiG.

A set E=e1,e2,…,eq be edge set of G and E=e1,e2,…,eq be set of cut vertices of G. The vertex set of lict graph is VnG=E(G)∪C(G). Consider E1∪C1, where E1⊆E and C1⊆C forms minimal a k-dominating set of n(G) such that E1∪C1+S≤V(G)+E(G).

Thus, γknG+γiG≤q+p.

Theorem 3.18: If G=p,qis any graph, then γknG≤diamG+β0G+△'G.

Proof. Case-I: If the graph has no cut vertex.

In this scenario, VnG=w1,w2,…,wqis the vertex set of nG with vertices corresponding to edges incident on vertices of G. Let D=ei|1≤i≤s be the edges lying on the longest path between two vertices u and v of G. Additionally, let A=u1,u2,…,ut be maximum independent set of G and B=ej|1≤j≤m be edges such that ej incident on uj, for all uj∈A. Set C=er|1≤r≤l consist of edges adjacent to an edge of maximum degree other than the element of D and B such that C≤△'(G). Consider D1∪B1∪C1, where D1⊆D, C1⊆C and B1⊆B forms minimal a k-dominating set of n(G), such that D1∪B1∪C1≤D+B+C.

This implies that, γknG≤diamG+β0G+△'G.

Case-II: If the graph has cut vertices.

In this case, VnG=w1,w2,…,wq+c be vertex set of nG with vertices of lict graph corresponding to edges and cut vertices of G. Let U=u1,u2,…,us contain vertices of lict graph corresponding to all cut vertices lying on the longest path between two vertices u and v of G such that |U| = diam(G). Additionally, let T=er|1≤r≤l be edges adjacent to an edge of maximum degree such that T=△'(G), and let S=ei|1≤i≤t contains edges such that ei incident on ui, for all ui∈A, where A=u1,u2,…,um is maximum independent set. Consider U1∪T1∪S1, where U1⊆U, T1⊆T and S1⊆S forms minimal a k-dominating set of n(G), such that U1∪T1∪S1≤U+T+S.

This implies that, γknG≤diamG+β0G+△'G.

Thus, it follows the result.

Theorem 3.19: If G=p,qis any graph and k is a positive integer with k≥2, then γknG≥β1G.

Proof. Let A=e1,e2,…,es be maximum subset of the edge set of G. Where no two edges from set A are adjacent. Therefore, A=β1G. A set VnG=w1,w2,…,wq+c be vertex set of nG with vertices correspond to edges and cut vertices of G.

For k≥2, if every vertex of VnG-A is adjacent to at least k vertices of A, then A is minimal k-dominating set of n(G). Otherwise, by adding vertices from VnG-A to set A, we can obtain a minimal k-dominating set of n(G).

Clearly, γknG≥β1G.

This completes the proof.

Theorem 3.20: If G=p,q be any graph, then γknG≤α1G+α0G+γtG.

Proof. Let VnG=w1,w2,…,wq+cbe vertex set of n(G) and members of VnG correspond to edges and cut vertices of G. Let A1=ei|1≤i≤s be minimal edge cover set of G such that A1=α1G. Also, let A2=ej|1≤j≤t contains edges of G incident on vertices of minimal vertex cover set of G such that A2=α0G. Additionally, let T=u1,u2,…,um be a minimal total dominating set of G, where cut vertices of G are members of T.

Consider, A⊆A1, A3⊆A2 and T1⊆T, then A∪A3∪T1 forms minimal a T k-dominating set of nGsuch that: A∪A3∪T1≤A1+A2+T.

Therefore, γknG≤α1G+α0G+γtG.

Theorem 3.21: For any graph G=p,q and a positive integer k. If k=△(n(G)) then γknG≥q+c-s, where s is number of vertices of degree △(n(G)).

Proof. The lict graph has q+c vertices, consisting q edges and c cut vertices. Let S=w1,w2,…,ws contains a vertices of degree △(n(G)). The proof is divided into two cases.

Case-I:S=1.

In this case, the result hold trivially.

Case-II:S>1,

Subcase-I.For k=△nG, if the members of S are adjacent to each other:

In this scenario, a k-dominating set contain S-1 vertices from S. We get required result.

Subcase-II. For k=△nG, if the members of S are not adjacent to each other:

In this case, there exists kmembers from V(n(G)) other than member of S adjacent to every vertex of S. we get, γknG≥q+c-s.

Therefore, the theorem holds true.

Theorem 3.22: If G=p,qbe any graph, then γknG≤γklG+c, where c is number of cut vertices of G.

Proof. Case-I: If the graph has no cut vertex:

In this case, the result holds because the lict graph coincides with line graph.

Case-II: If the graph has cut vertices:

Let D be k dominating set of L(G) and S be set of cut vertices of G, which are not incident with any edge of D. For every cut vertex u∈S, consider exactly one edge in E1, where E1=ej∈E(G)|ejincidentwithuandej∈N(D) such that, E1≤S. This implies that D∪E1 forms minimal k-dominating set of lict graph of G. It follows that γknG≤γklG+S.

Therefore, γknG≤γklG+c.

Theorem 3.23: If G=p,q be any graph and a positive integer k with k≤δG, then γknG≤p

Proof. Let V(G)=u1,u2,…,up be vertex set of G and VnG=w1,w2,…,wq+cbe the vertex set of lict graph of G. We consider subset D1=w1,w2,…,wq+c of VnG, which corresponds to edges incident on vertices of G and cut vertices of G.

For k≤δ(G), there exists D2⊆D1is minimal k-dominating set of lict graph of G such that D2≤VG=p.

Therefore, γknG≤p, and the proof is complete.

Theorem 3.24: If G=p,qbe any graph and a positive integer k with k≤δG, then γknG≤q.

Proof. Let VnG=w1,w2,…,wq+c be the vertex set of lict graph of G. Consider the subset E=w1,w2,…,wq of VnG with vertices of E corresponding to edges incident on vertices of G such that E=q.

For k≤δG, then there exists a possible E1⊆E that is minimal k-dominating set of lict graph of G such that E1≤E.

Therefore, γknG≤q.

Theorem 3.25: If G=p,qbe any graph and a positive integer k with k≥△G, then γknG≥△G.

Proof. Since for every k, and k≥△n(G), it follows that γknG≥△n(G).

For any graph △nG≥△(G).

Therefore, γknG≥△G.

Theorem 3.26: If G=p,q be any graph with p≥2, then γknG+γ'G≤q+γctG+γsenG.

Proof. Using Theorem 2.3, Theorem 2.5 and Theorem 3.12, we get the required result.

Theorem 3.27: If G=p,q be any graph with p≥2, then γknG+γ'G≤q+γctG+γtnG.

Proof. Using Theorem 2.3, Theorem 2.6 and Theorem 3.12, we get the required result.

<bold id="s-6715b72e2863">Conclusion</bold>

This study has computed the results regarding k-domination in lict graph. The study involves deriving theorems specifically related to lict k-domination and relationships with various other domination parameters. Furthermore, this investigation derivation of lict k-domination number, denoted as γknG, considering factors such as vertices, edges, and various other parameters of graph G.

<bold id="s-031df72fa707">Acknowledgement</bold>

The research is funded by Council of Scientific and Industrial Research Human Resource Development Group (CSIR-HRDG). The authors would like to acknowledge the reviewers for their valuable suggestions for improvement and the author also thankful to CSIR-HRDG providing financial support under grant no. grant no. 09/0990(11223)/2021-EMR-I.

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