▁representation . ▁ ▁In ▁the ▁other ▁direction , ▁from ▁a ▁rib bon ▁graph ▁one ▁may ▁find ▁the ▁faces ▁of ▁its ▁corresponding ▁embedding ▁as ▁the ▁components ▁of ▁the ▁boundary ▁of ▁the ▁topological ▁surface ▁formed ▁by ▁the ▁rib bon ▁graph . ▁One ▁may ▁recover ▁the ▁surface ▁itself ▁by ▁gl uing ▁a ▁topological ▁disk ▁to ▁the ▁rib bon ▁graph ▁along ▁each ▁boundary ▁component . ▁The ▁partition ▁of ▁the ▁surface ▁into ▁vertex ▁dis ks , ▁edge ▁dis ks , ▁and ▁face ▁dis ks ▁given ▁by ▁the ▁rib bon ▁graph ▁and ▁this ▁gl uing ▁process ▁is ▁a ▁different ▁but ▁related ▁representation ▁of ▁the ▁embedding ▁called ▁a ▁' band ▁decomposition . ▁ ▁The ▁surface ▁onto ▁which ▁the ▁graph ▁is ▁embedded ▁may ▁be ▁determined ▁by ▁whether ▁it ▁is ▁orient able ▁( true ▁if ▁any ▁cycle ▁in ▁the ▁graph ▁has ▁an ▁odd ▁number ▁of ▁tw ists ) ▁and ▁by ▁its ▁E uler ▁characteristic . ▁ ▁The ▁embed d ings ▁that ▁can ▁be ▁represented ▁by ▁rib bon ▁graphs ▁are ▁the ▁ones ▁in ▁which ▁a ▁graph ▁is ▁embedded ▁onto ▁a ▁ 2 - man if old ▁( without ▁boundary ) ▁and ▁in ▁which ▁each ▁face ▁of ▁the ▁embedding ▁is ▁a ▁topological ▁disk . ▁ ▁Equ ival ence ▁Two ▁rib bon ▁graph ▁representations ▁are ▁said ▁to ▁be ▁equivalent ▁( and ▁define ▁home omorphic ▁graph ▁embed d ings ) ▁if ▁they ▁are ▁related ▁to ▁each ▁other ▁that ▁a ▁home omorphism ▁of ▁the ▁topological ▁space ▁formed ▁by ▁the ▁union ▁of ▁the ▁vertex ▁dis ks ▁and ▁edge ▁rect angles ▁that ▁pres erves ▁the ▁identification ▁of ▁these ▁features