Največje neodvisne množice v direktnih produktih ciklov in dreves s poljubnimi grafi


The direct product of graphs ▫G=(V(G),E(G))G = (V(G),E(G))▫ and ▫H=(V(H),E(H))H = (V(H),E(H))▫ is the graph, denoted as ▫GtimesHG times H▫, with vertex set ▫V(GtimesH)=V(G)timesV(H)V(G times H) = V(G )times V(H)▫, where vertices ▫(x1,y1)(x_1,y_1)▫ and ▫(x2,y2)(x_2,y_2)▫ are adjacent in ▫GtimesHG times H▫ if ▫x1x2inE(G)x_1x_2 in E(G)▫ and ▫y1y2inE(H)y_1y_2 in E(H)▫. Let ▫nn▫ be odd and ▫mm▫ even. We prove that every maximum independent set in ▫PntimesGP_n times G▫, respectively ▫CmtimesGC_m times G▫, is of the form ▫(AtimesC)cup(BtimesD)(A times C) cup (B times D)▫, where ▫CC▫ and ▫DD▫ are nonadjacent in ▫GG▫, and ▫AcupBA cup B▫ is the bipartition of ▫PnP_n▫ respectively ▫CmC_m▫. We also give a characterization of maximum independent subsets of ▫PntimesGP_n times G▫ for every even ▫nn▫ and discuss the structure of maximum independent sets in ▫TtimesGT times G▫ where ▫TT▫ is a tree.Dokažemo, da je vsaka največja neodvisna množica v direktnem produktu sodega cikla ali lihe poti s poljubnim grafom GG unija množic (AtimesB(A times B) in (CtimesD(C times D), kjer sta AA in CC podmnožici cikla oziroma poti, ter BB in DD podmnožici grafa GG. Prav tako diskutiramo strukturo največjih neodvisnih množic v direktnih produktih dreves s poljubnimi grafi

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