▁lattice ▁of ▁dimensions ▁linear ▁in ▁the ▁graph ▁sizes . ▁Similarly ▁defined ▁layout s ▁also ▁work , ▁with ▁larger ▁but ▁still ▁linear ▁grid ▁sizes , ▁when ▁both ▁graphs ▁are ▁c ater p ill ars ▁or ▁when ▁both ▁are ▁cycle ▁graphs . ▁A ▁simultane ous ▁embedding ▁in ▁a ▁grid ▁of ▁linear ▁dimensions ▁is ▁also ▁possible ▁for ▁any ▁number ▁of ▁graphs ▁that ▁are ▁all ▁stars . ▁Other ▁pairs ▁of ▁graph ▁types ▁that ▁always ▁admit ▁a ▁simultane ous ▁embedding , ▁but ▁that ▁might ▁need ▁larger ▁grid ▁sizes , ▁include ▁a ▁wheel ▁graph ▁and ▁a ▁cycle ▁graph , ▁a ▁tree ▁and ▁a ▁matching , ▁or ▁a ▁pair ▁of ▁graphs ▁both ▁of ▁which ▁have ▁maximum ▁degree ▁two . ▁However , ▁pairs ▁of ▁plan ar ▁graphs ▁and ▁a ▁matching , ▁or ▁of ▁a ▁tree ▁and ▁a ▁path ▁exist , ▁that ▁have ▁no ▁simultane ous ▁geometric ▁embedding . ▁ ▁Test ing ▁whether ▁two ▁graphs ▁admit ▁a ▁simultane ous ▁geometric ▁embedding ▁is ▁N P - hard . ▁More ▁precisely , ▁it ▁is ▁complete ▁for ▁the ▁exist ential ▁theory ▁of ▁the ▁re als . ▁The ▁proof ▁of ▁this ▁result ▁also ▁implies ▁that ▁for ▁some ▁pairs ▁of ▁graphs ▁that ▁have ▁simultane ous ▁geometric ▁embed d ings , ▁the ▁smallest ▁grid ▁on ▁which ▁they ▁can ▁be ▁drawn ▁has ▁doub ly ▁exponential ▁size . ▁ ▁When ▁a ▁simultane ous ▁geometric ▁embedding ▁exists , ▁it ▁automatically ▁is ▁also ▁a ▁simultane ous ▁embedding ▁with ▁fixed ▁edges . ▁ ▁F ixed ▁edges ▁In ▁simultane ous ▁embedding ▁with ▁fixed ▁edges , ▁curves ▁or ▁b ends ▁are ▁allowed ▁in ▁the ▁edges