Povezanost posplošenih Fibonaccjevih in Lucasovih kock

If ▫$f$▫ is a binary word and ▫$d$▫ a positive integer, then the generalized Fibonacci cube ▫$Q_d(f)$▫ is the graph obtained from the ▫$d$▫-cube ▫$Q_d$▫ by removing all the vertices that contain ▫$f$▫ as a factor, while the generalized Lucas cube ▫$Q_d(stackrel{leftharpoondown}{f})$▫ is the graph obtained from ▫$Q_d$▫ by removing all the vertices that have a circulation containing ▫$f$▫ as a factor. The Fibonacci cube ▫$Gamma_d$▫ and the Lucas cube ▫$Lambda_d$▫ are the graphs ▫$Q_d({11})$▫ and ▫$Q_d(stackrel{leftharpoondown}{11})$▫, respectively. It is proved that the connectivity and the edge-connectivity of ▫$Gamma_d$▫ as well as of ▫$Lambda_d$▫ are equal to ▫$leftlfloor frac{d+2}{3}rightrfloor$▫. Connected generalized Lucas cubes are characterized and generalized Fibonacci cubes are proved to be 2-connected. It is asked whether the connectivity equals minimum degree also for all generalized Fibonacci/Lucas cubes. It was checked by computer that the answer is positive for all ▫$f$▫ and all ▫$d le9$▫.Za binarni niz ▫$f$▫ in naravno število ▫$d$▫ pravimo, da je ▫$Q_d(f)$▫ posplošena Fibonaccijeva kocka, pri čemer je to graph pridobljen iz ▫$d$▫-kocke ob odstranitvi vseh vozlišč ki vsebujejo ▫$f$▫ kot podniz. Na podoben način vpeljemo posplošeno Lucasovo kocko ▫$Q_d(stackrel{leftharpoondown}{f})$▫ kot graf pridobljen iz ▫$Q_d$▫ z odstranitvijo vozlišč ▫$d$▫-kocke, ki vsebujejo ▫$f$▫ kot cirkularen podniz. S to notoacijo sta Fibonaccijeva kocka ▫$Gamma_d$▫ in Lucasova kocka ▫$Lambda_d$▫ grafa ▫$Q_d(11)$▫ in ▫$Q_d(stackrel{leftharpoondown}{11})$▫. V članku je dokazano da je vozliščna in povezavna povezanost za ▫$Gamma_d$▫ in ▫$Lambda_d$▫ enaka ▫$leftlfloor frac{d+2}{3}rightrfloor$▫. Postavljeno je tudi vprašanje, ali je povezanost posplošenih Fibonaccijevih in Lucasovih kock enaka minimalni stopnji. Slednje je bilo preverjeno z računalnikom za ▫$d le 9$▫ in vse možne ▫$f$▫