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The edge general position problem
Manuel, Paul ; Prabha, R. ; Klavžar, Sandi
Problem splošne lege je iskanje največje množice ▫$S$▫ vozlišč grafa ▫$G$▫, tako da nobena trojica iz ▫$S$▫ ne leži na skupni najkrajši poti. Taka množica se imenuje ▫${\rm gp}$▫-množica v ▫$G$▫, ...njena kardinalnost je gp-število, ▫${\rm gp}(G)$▫, v ▫$G$▫. V tem članku je uveden problem splošne lege za povezave. Število splošne lege za povezave, ▫${\rm gp_{e}}(G)$▫, je velikost največje množice povezav v ▫$G$▫, ki so v splošni legi. Za ▫$r$▫-dimenzionalno hiperkocko ▫$Q_r$▫ je dokazano, da je ▫${\rm gp_{e}}(Q_r) = 2^r$▫, za poljubno drevo ▫$T$▫ pa je dokazano, da je ▫${\rm gp_{e}}(T)$▫ število njegovih listov. Vrednost ▫${\rm gp_{e}}(P_r\, \square\, P_s)$▫ je določena za vsak ▫$r,s\ge 2$▫. Za izpeljavo teh rezultatov se uporablja teorija delnih kock. Ob tem je obravnavana tudi Mulderjeva meta-domneva o medianskih grafih.
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