▁a ▁copy ▁of ▁the ▁unit ▁interval ▁, ▁where ▁ ▁is ▁identified ▁with ▁the ▁point ▁associated ▁to ▁ ▁and ▁ ▁with ▁the ▁point ▁associated ▁to ▁. ▁That ▁is , ▁as ▁topological ▁spaces , ▁graphs ▁are ▁exactly ▁the ▁simp lic ial ▁ 1 - comple xes ▁and ▁also ▁exactly ▁the ▁one - dimensional ▁C W ▁complex es . ▁ ▁Thus , ▁in ▁particular , ▁it ▁be ars ▁the ▁quot ient ▁topology ▁of ▁the ▁set ▁ ▁under ▁the ▁quot ient ▁map ▁used ▁for ▁gl uing . ▁Here ▁ ▁is ▁the ▁ 0 - ske leton ▁( cons isting ▁one ▁point ▁for ▁each ▁vertex ▁), ▁ ▁are ▁the ▁intervals ▁(" closed ▁one - dimensional ▁unit ▁balls ") ▁gl ued ▁to ▁it , ▁one ▁for ▁each ▁edge ▁, ▁and ▁ ▁is ▁the ▁dis joint ▁union . ▁ ▁The ▁topology ▁on ▁this ▁space ▁is ▁called ▁the ▁graph ▁topology . ▁ ▁Sub graph s ▁and ▁- trees ▁▁ ▁A ▁sub graph ▁of ▁a ▁graph ▁ ▁is ▁a ▁sub space ▁ ▁which ▁is ▁also ▁a ▁graph ▁and ▁whose ▁nodes ▁are ▁all ▁contained ▁in ▁the ▁ 0 - ske leton ▁of ▁. ▁ ▁is ▁a ▁sub graph ▁if ▁and ▁only ▁if ▁it ▁consists ▁of ▁vertices ▁and ▁edges ▁from ▁ ▁and ▁is ▁closed . ▁ ▁A ▁sub graph ▁ ▁is ▁called ▁a ▁tree ▁if f ▁it ▁is ▁contract ible ▁as ▁a ▁topological ▁space . ▁ ▁Properties ▁▁▁ ▁Every ▁connected ▁graph ▁ ▁contains ▁at ▁least ▁one ▁maximal ▁tree ▁, ▁that ▁is , ▁a ▁tree ▁that ▁is ▁maximal ▁with ▁respect ▁to ▁the ▁order ▁induced ▁by