▁partition ed ▁into ▁two ▁sub graph s ▁that ▁have ▁a ▁simultane ous ▁embedding , ▁and ▁the ▁geometric ▁thick ness ▁is ▁two , ▁if ▁the ▁edges ▁can ▁be ▁partition ed ▁into ▁two ▁sub graph s ▁with ▁simultane ous ▁geometric ▁embedding . ▁ ▁Ge ometric ▁ ▁In ▁simultane ous ▁geometric ▁embedding ▁ ▁each ▁graph ▁must ▁be ▁drawn ▁as ▁a ▁plan ar ▁graph ▁with ▁line ▁segments ▁representing ▁its ▁edges ▁rather ▁than ▁more ▁complex ▁curves ▁restrict ing ▁the ▁two ▁given ▁graphs ▁to ▁sub classes ▁of ▁the ▁plan ar ▁graphs . ▁Many ▁results ▁on ▁simultane ous ▁geometric ▁embedding ▁are ▁based ▁on ▁the ▁idea ▁that ▁the ▁Cart esian ▁coordinates ▁of ▁the ▁two ▁given ▁graphs ' ▁vertices ▁can ▁be ▁derived ▁from ▁properties ▁of ▁the ▁two ▁graphs . ▁One ▁of ▁the ▁most ▁basic ▁results ▁of ▁this ▁type ▁is ▁the ▁fact ▁that ▁any ▁two ▁path ▁graphs ▁on ▁the ▁same ▁vertex ▁set ▁always ▁have ▁a ▁simultane ous ▁embedding . ▁To ▁find ▁such ▁an ▁embedding , ▁one ▁can ▁use ▁the ▁position ▁of ▁a ▁vertex ▁in ▁the ▁first ▁path ▁as ▁its ▁x - coordinate , ▁and ▁the ▁position ▁of ▁the ▁same ▁vertex ▁in ▁the ▁second ▁path ▁as ▁its ▁y - coordinate . ▁In ▁this ▁way , ▁the ▁first ▁path ▁will ▁be ▁drawn ▁as ▁an ▁x - mon ot one ▁poly line , ▁a ▁type ▁of ▁curve ▁that ▁is ▁automatically ▁non - self - cross ing , ▁and ▁the ▁second ▁path ▁will ▁similarly ▁be ▁drawn ▁as ▁a ▁y - mon ot one ▁poly line . ▁ ▁This ▁type ▁of ▁drawing ▁places ▁the ▁vertices ▁in ▁an ▁integer