 oai:dk.um.si:IzpisGradiva.php?id=65476

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

## Abstract

The direct product of graphs ▫$G = (V(G),E(G))$▫ and ▫$H = (V(H),E(H))$▫ is the graph, denoted as ▫$G times H$▫, with vertex set ▫$V(G times H) = V(G )times V(H)$▫, where vertices ▫$(x_1,y_1)$▫ and ▫$(x_2,y_2)$▫ are adjacent in ▫$G times H$▫ if ▫$x_1x_2 in E(G)$▫ and ▫$y_1y_2 in E(H)$▫. Let ▫$n$▫ be odd and ▫$m$▫ even. We prove that every maximum independent set in ▫$P_n times G$▫, respectively ▫$C_m times G$▫, is of the form ▫$(A times C) cup (B times D)$▫, where ▫$C$▫ and ▫$D$▫ are nonadjacent in ▫$G$▫, and ▫$A cup B$▫ is the bipartition of ▫$P_n$▫ respectively ▫$C_m$▫. We also give a characterization of maximum independent subsets of ▫$P_n times G$▫ for every even ▫$n$▫ and discuss the structure of maximum independent sets in ▫$T times G$▫ where ▫$T$▫ is a tree.Dokažemo, da je vsaka največja neodvisna množica v direktnem produktu sodega cikla ali lihe poti s poljubnim grafom $G$ unija množic $(A times B$) in $(C times D$), kjer sta $A$ in $C$ podmnožici cikla oziroma poti, ter $B$ in $D$ podmnožici grafa $G$. Prav tako diskutiramo strukturo največjih neodvisnih množic v direktnih produktih dreves s poljubnimi grafi

## Full text ### Digital library of University of Maribor

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This paper was published in Digital library of University of Maribor.

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