Dimensi Partisi pada Graf Payung

Abstract

Let G(V,E) be aconnectedgraphwiththevertexset V and thesetofedges E. GivenapartitionΠof V (G) with k component classintheformΠ = {L1, L2, · ·· , Lk}. The representationofthepoint t withrespectto Π is definedasavectorwith k components canbewrittenintheformof r(t|Π) =(d(t, L1), d(t, L2), · ·· , d(t, Lk)), where k is apositiveinteger.Foragraph G is connectedandasubsetof L ⊂ V (G), the partition Π is calledthedistinguishingpartitionofthegraph G if allrepresentations of thepoint t ∈ V (G) aredifferentwithrespectto Π. The smallestpositiveinteger k is thepartitiondimensionofthegraph G denotedby pd(G). In thisresearch,we will determinethepartitiondimensionofumbrellagraph Um,n(1) and Um,n(2). Graph Um,n(1) is agraphmergedfromawheelgraph W1,n and path Pn. Graph Um,n(2) is a graphmergedfromafangraph F1,n and path Pn.