▁Category : J apan ese ▁male ▁compos ers ▁Category : J apan ese ▁rock ▁keyboard ists ▁Category : K an ag awa ▁University ▁al umn i ▁Category : L iving ▁people ▁Category : M ale ▁film ▁score ▁compos ers ▁Category : Mus icians ▁from ▁K ō chi ▁Pref ect ure ▁Category : Pe ople ▁from ▁K ō chi , ▁K ō chi ▁Category : Progress ive ▁rock ▁keyboard ists ▁Category : S quare ▁En ix ▁people ▁Category : Video ▁game ▁compos ers <0x0A> </s> ▁In ▁the ▁mathematics ▁of ▁struct ural ▁rig id ity , ▁a ▁rig id ity ▁mat roid ▁is ▁a ▁mat roid ▁that ▁describes ▁the ▁number ▁of ▁degrees ▁of ▁freedom ▁of ▁an ▁und irect ed ▁graph ▁with ▁rig id ▁edges ▁of ▁fixed ▁lengths , ▁embedded ▁into ▁E uclidean ▁space . ▁In ▁a ▁rig id ity ▁mat roid ▁for ▁a ▁graph ▁with ▁n ▁vertices ▁in ▁d - dimensional ▁space , ▁a ▁set ▁of ▁edges ▁that ▁defines ▁a ▁sub graph ▁with ▁k ▁degrees ▁of ▁freedom ▁has ▁mat roid ▁rank ▁d n   −   k . ▁A ▁set ▁of ▁edges ▁is ▁independent ▁if ▁and ▁only ▁if , ▁for ▁every ▁edge ▁in ▁the ▁set , ▁removing ▁the ▁edge ▁would ▁increase ▁the ▁number ▁of ▁degrees ▁of ▁freedom ▁of ▁the ▁remaining ▁sub graph . ▁ ▁Definition ▁A ▁framework ▁is ▁an ▁und irect ed ▁graph , ▁embedded ▁into ▁d - dimensional ▁E uclidean ▁space ▁by ▁providing ▁a ▁d - tuple ▁of ▁Cart esian ▁coordinates ▁for ▁each ▁vertex ▁of ▁the ▁graph . ▁From ▁a ▁framework ▁with ▁n ▁vertices