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Chứng minh: Chúng tôi chứng minh định lý này bằng cách mâu thuẫn. Giả sử G là bị ngắt kết nối và v và u là hai đỉnh bị ngắt kết nối trong G. Giả sử disG (v, u) = k + 1 1 và (v, v1, v2,, Vk, u) là một đường đi ngắn nhất giữa các đỉnh v và u trong G. | 20.3 FORMATION OF A CONNECTED DOMINATING SET 433 Theorem 2 The induced graph G G K is a connected graph. Proof We prove this theorem by contradiction. Assume that G is disconnected and V and u are two disconnected vertices in G . Assume disG y u k 1 1 and v v1 v2 . . vk u is a shortest path between vertices V and u in G. Clearly all V-1 V2 . vk are distinct and among them there is at least one Vi such that m Vi F otherwise V and u are connected in G . On the other hand the two adjacent vertices of Vi Vi_Ị and vi 1 are not connected in G otherwise v Vị v2 . VoìM isnotashortestpath.oniereíore T based on the marking process. This brings a contradiction. The next theorem shows that except for source and destination vertices all intermediate vcrticd cno ashortestpath arecontained mthedominating set derivedfrom themark-ing procoss. Tl eoumt The sPortestpatgOeVween anylwo nodesdoes ivtsinclrdi anynongataway nede-snn intermeSiaVenoda. Pooof Wcpr vetlnsdieorem alsoby kontoadikOngd.Asvnme that a shortest path between two vertices V and u includes a nongateway node Vi as an intermediate node in other woads kiis padintm Oarc o rescn cdds Av. d _b d P 1 . taa-Wetabaa dievortex diaCpi ccedecct os nop o athosntsi iimilprly thevertexthal foilows a ot -bkOndorttoi Because vertex is a nongateway node i.e. m Vi F there must be a connection between Vị-i and VM. Therefore a shorter path between V and u can be found as v . Vi3 Vti .- id.i luoc Piodiil. tlee iigiooiasrmnpl on oe Since the problem of determining a minimum connected dominating set of a given con-nevted cip 1i s NPtCompleh tOo PhcmecCod dominhtino sptderivodfrpmthe P diCiao orocass idnomok-y nonm om.ln tome canoodle r tsohalìtd mtaaiiaolet ìsIcóoú Ìi ó.e. ik naoK Ttt.Forexample aay vtrtee-tymm etric graph win generated rovial domtnadooset uoii - .-. sedmorhingprosesr. HawovoO themaremgprocassis o H-Pu dt r enad hdtwiraiest network wkooetOo cooretpoodlng grape teiidnCoionnasiTof locaiókddlottaks or olioe -.rg. asp .
