Edge metric dimensions via hierarchical product and integer linear programming
Klavžar, Sandi ; Tavakoli, Mostafa
Če je ▫$S=\{v_1,\ldots, v_k\}$▫ urejena podmnožica vozlišč povezanega grafa ▫$G$▫ in je ▫$e$▫ povezava grafa ▫$G$▫, potem je vektor ▫$r_G(e|S) = (d_G(v_1,e), \ldots, d_G(v_k,e))$▫ povezavna metrična ...▫$S$▫-predstavitev ▫$e$▫. Če imajo vozlišča ▫$G$▫ paroma različne povezavne metrične ▫$S$▫-predstavitve, potem je ▫$S$▫ povezavni metrični generator za ▫$G$▫. Moč najmanjšega takega generatorja je povezavna metrična dimenzija ▫${\rm edim}(G)$▫ grafa ▫$G$▫. Dokazana je splošna ostra zgornja meja povezavne metrične dimenzije hierarhičnih produktov ▫$G(U) \sqcap H$▫. Natančna formula je izpeljana za primer, ko je ▫$|U| = 1$▫. Predlagan je model celoštevilskega linearnega programiranja za izračun povezavne metrične dimenzije. Navedenih je več primerov, ki demonstrirajo, kako je mogoče ti dve metodi uporabiti za določitev povezavne metrične dimenzije nekaterih uporabnih grafov.
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