/ / / / / / / figure depth-first tree for previous graphwith num and low back edge this third method is succinctly described with recursive call since we need to evaluate low for all the children of before we can evaluate low( )this is postorder traversal for any edge (vw)we can tell whether it is tree edge or back edge merely by checking num(vand num(wthusit is easy to compute low( )we merely scan down ' adjacency listapply the proper ruleand keep track of the minimum doing all the computation takes (| | |time all that is left to do is to use this information to find articulation points the root is an articulation point if and only if it has more than one childbecause if it has two childrenremoving the root disconnects nodes in different subtreesand if it has only one childremoving the root merely disconnects the root any other vertex is an articulation point if and only if has some child such that low( >num(vnotice that this condition is always satisfied at the roothence the need for special test the if part of the proof is clear when we examine the articulation points that the algorithm determinesnamelyc and has child eand low( >num( )since both are thusthere is only one way for to get to any node above dand that is by going through similarlyc is an articulation pointbecause low( >num(cto prove that this algorithm is correctone must show that the only if part of the assertion is true (that isthis finds all articulation pointswe leave this as an exercise as second examplewe show (fig the result of applying this algorithm on the same graphstarting the depth-first search at