▁character ization ▁of ▁the ▁uniqu ely ▁realiz able ▁frameworks . ▁The ▁conject ure ▁is ▁true ▁for ▁one ▁and ▁two ▁dimensions ; ▁in ▁the ▁one - dimensional ▁case , ▁for ▁instance , ▁a ▁graph ▁is ▁uniqu ely ▁realiz able ▁if ▁and ▁only ▁if ▁it ▁is ▁connected ▁and ▁brid g eless . ▁However , ▁Hen rick son ' s ▁conject ure ▁is ▁false ▁for ▁three ▁or ▁more ▁dimensions . ▁For ▁frameworks ▁that ▁are ▁not ▁generic , ▁it ▁is ▁N P - hard ▁to ▁determine ▁whether ▁a ▁given ▁framework ▁is ▁uniqu ely ▁realiz able . ▁ ▁Rel ation ▁to ▁sp ars ity ▁ ▁define ▁a ▁graph ▁as ▁being ▁- s parse ▁if ▁every ▁non empty ▁sub graph ▁with ▁ ▁vertices ▁has ▁at ▁most ▁ ▁edges , ▁and ▁- t ight ▁if ▁it ▁is ▁- s parse ▁and ▁has ▁exactly ▁ ▁edges . ▁From ▁the ▁consideration ▁of ▁loads ▁and ▁str esses ▁it ▁can ▁be ▁seen ▁that ▁a ▁set ▁of ▁edges ▁that ▁is ▁independent ▁in ▁the ▁rig id ity ▁mat roid ▁forms ▁a ▁- s parse ▁graph , ▁for ▁if ▁not ▁there ▁would ▁exist ▁a ▁sub graph ▁whose ▁number ▁of ▁edges ▁would ▁exceed ▁the ▁dimension ▁of ▁its ▁space ▁of ▁equilibrium ▁loads , ▁from ▁which ▁it ▁follows ▁that ▁it ▁would ▁have ▁a ▁self - str ess . ▁By ▁similar ▁reasoning , ▁a ▁set ▁of ▁edges ▁that ▁is ▁both ▁independent ▁and ▁rig id ▁forms ▁a ▁- t ight ▁graph . ▁For ▁instance , ▁in ▁one ▁dimension , ▁the ▁independent ▁sets ▁form ▁the ▁edge ▁sets ▁of ▁for ests , ▁( 1