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On graphs with the maximum edge metric dimension
Zhu, Enqiang ...
Povezavni metrični generator povezanega grafa ▫$G$▫ je podmnožica vozlišč ▫$S$▫, za katero imata poljubni dve različni povezavi grafa ▫$G$▫ različno razdaljo do nekega vozlišča iz ▫$S$▫; pri tem je ...razdalja med vozliščem ▫$v$▫ in povezavo ▫$e$▫ definirana kot minimum razdalj med ▫$v$▫ in krajiščema povezave ▫$e$▫ v grafu ▫$G$▫. Moč najmanjšega povezavnega metričnega generatorja je povezavna metrična dimenzija, označimo jo z ▫$\dim_e(G)$▫. Za poljuben graf ▫$G$▫ reda ▫$n$▫ je ▫$1 \leq \dim_e(G) \leq n-1$▫. Graf, katerega povezavna metrična dimenzija doseže zgornjo mejo, imenujemo zvrhan. V tem članku so podani karakterizacija strukture zvrhanih grafov ter zadostni in potrebni pogoji za to, da je graf zvrhan. Z uporabo teh rezultatov je podan algoritem, ki v času ▫$O(n^3)$▫ preveri, ali je podani graf reda ▫$n$▫ zvrhan ali ni. Opisana in proučevana je tudi zanimiva družina zvrhanih grafov, katerih nadgrafi dobljeni z dodajanjem ene povezave niso zvrhani.
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