DIMENSI PARTISI GRAF THORN DARI GRAF RODA W3 DAN W4

Abstract

Let ðº = (ð‘‰, ð¸) be a connected graph and 𑆠⊆ ð‘‰(ðº). For a vertex v ∈ V(G) and an ordered k-partition Π = {ð‘†1 , ð‘†2 , … , ð‘†ð‘˜ } of ð‘‰(ðº), the representation of v with respect to Π is the k-vector ð‘Ÿ(ð‘£|ð›± = (ð‘‘(ð‘£, ð‘†1), ð‘‘(ð‘£, ð‘†2), . . . , ð‘‘(ð‘£, ð‘†ð‘˜)), where d(v,Si) denotes the distance between v and Si. The k-partition Π is said to be resolving if for every two vertices ð‘¢, 𑣠 ð‘‰(ðº), the representation ð‘Ÿ(ð‘¢|П)  ð‘Ÿ(ð‘£|Π). The minimum k for which there is a resolving k-partition of ð‘‰(ðº) is called the partition dimension of ðº, denoted by ð‘ð‘‘(ðº). The wheel graph ð‘Šð‘› ð‘œð‘› ð‘› + 1 vertices with ð‘‰(ð‘Šð‘›) = {ð‘£0, ð‘£1, . . . , ð‘£ð‘›}. Let ð‘™2 ,ð‘™2 ,… ,ð‘™ð‘›be non-negative integers, ð‘™ð‘– ≥ 1, for 𑖠 {0,1,2, . . . , ð‘›}. The thorn graph of the graph Wn, with parameters ð‘™0 ,ð‘™1 ,… ,ð‘™ð‘› is obtained by attaching li new vertices of degree one to the vertex vi of the graph Wn. The thorn graph is denoted by ð‘‡â„Ž(ð‘Šð‘›,ð‘™0 ,ð‘™1 ,… ,ð‘™ð‘›). In this paper we give the upper bounds for the partition dimension of ð‘Š3 and ð‘Š4 denoted by ð‘ð‘‘(ð‘‡â„Ž(ð‘Š3 ,ð‘™0 ,ð‘™1 ,ð‘™2 ,ð‘™3 )) and ð‘ð‘‘(ð‘‡â„Ž(ð‘Š4 ,ð‘™0 ,ð‘™1 ,ð‘™2 ,ð‘™3 ,ð‘™4 )). Keywords : Partition Dimension, Resolving Partition, Thorn Graph, Wheel Graph.