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Retracts of products of chordal graphs
Brešar, Boštjan ...
V članku karakteriziramo tiste grafe, ki so retrakti kartezičnih produktov tetivnih grafov. Dokažemo, da so to natanko šibko modularni grafi, ki ne vsebujejo grafa ▫$K_{2,3}$▫, 4-kolesa brez ene ...napere ▫$W_4^-$▫ in ▫$k$▫-koles ▫$W_k$▫ (za ▫$k \ge 4$▫) kot induciranih podgrafov. Dokažemo tudi, da so ti grafi natanko grafi amalgamov kletk, kot sta jih vpeljala Brešar in Tepeh Horvat (Cage-amalgamation graphs, a common generalization of chordal and median graphs, Eur. J. Combin. 30 (2009), 1071-1081); s tem rešimo odprto vprašanje, ki sta ga zastavila avtorja. Nazadnje dokažemo, da ob zamenjavi vse produktov klik v grafu ▫$G$▫ s produkti Evklidskih simpleksov, dobimo poliedrski celični kompleks, ki tvori CAT(0) prostor, če ga opremimo z intrinzično Evklidsko metriko. To posploši podobne rezultate o medianskih grafih kot retrakti hiperkock (produkti povezav) in medianskih grafih kot 1-skeletih CAT(0) kompleksov kock.
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