isions ▁( part itions ▁of ▁a ▁rect angular ▁sub div ision ▁into ▁smaller ▁rect angles , ▁with ▁three ▁rect angles ▁meeting ▁at ▁every ▁vertex ) ▁may ▁be ▁described ▁combin atorial ly ▁by ▁a ▁" regular ▁label ing ", ▁a ▁two - color ing ▁of ▁the ▁edges ▁of ▁a ▁tri ang ulation ▁dual ▁to ▁the ▁sub div ision , ▁with ▁the ▁constraint ▁that ▁the ▁edges ▁incident ▁to ▁each ▁vertex ▁form ▁four ▁cont igu ous ▁subsequ ences , ▁within ▁each ▁of ▁which ▁the ▁colors ▁are ▁the ▁same . ▁This ▁label ing ▁is ▁dual ▁to ▁a ▁color ing ▁of ▁the ▁rect angular ▁sub div ision ▁itself ▁in ▁which ▁the ▁vertical ▁edges ▁have ▁one ▁color ▁and ▁the ▁horizontal ▁edges ▁have ▁the ▁other ▁color . ▁Similar ▁local ▁constraints ▁on ▁the ▁order ▁in ▁which ▁colored ▁edges ▁may ▁appear ▁around ▁a ▁vertex ▁may ▁also ▁be ▁used ▁to ▁encode ▁straight - line ▁grid ▁embed d ings ▁of ▁plan ar ▁graphs ▁and ▁three - dimensional ▁poly hed ra ▁with ▁axis - parallel ▁sides . ▁For ▁each ▁of ▁these ▁three ▁types ▁of ▁regular ▁label ings , ▁the ▁set ▁of ▁regular ▁label ings ▁of ▁a ▁fixed ▁graph ▁forms ▁a ▁distrib utive ▁lattice ▁that ▁may ▁be ▁used ▁to ▁quickly ▁list ▁all ▁geometric ▁structures ▁based ▁on ▁the ▁same ▁graph ▁( such ▁as ▁all ▁axis - parallel ▁poly hed ra ▁having ▁the ▁same ▁ske leton ) ▁or ▁to ▁find ▁structures ▁satisfying ▁additional ▁constraints . ▁ ▁A ▁determin istic ▁finite ▁automat on ▁may ▁be ▁interpreted ▁as ▁a ▁directed ▁graph ▁in ▁which ▁each ▁vertex ▁has ▁the ▁same ▁out -