Question

Is there an algorithm (or a sequence of algorithms) to find, given a generic graph structure G=(V,E) with no notion of parent node, leaf node and child node but only neighboordhood relations:

1) If G it is a tree or not (is it sufficient to check |V| = |E|+1?)

2) If the graph is actually a tree, the leaves and the center of it? (i.e the node of the graph which minimizes the tree depth)

Thanks

Answer 1

If the “center” of the tree is defined as “the node of the graph which minimizes the tree depth”, there’s an easier way to find it than finding the diameter.

d[] = degrees of all nodes
que = { leaves, i.e  i that d[i]==1}
while len(que) > 1:
  i=que.pop_front
  d[i]--
  for j in neighbors[i]:
    if d[j] > 0:
      d[j]--
      if d[j] == 1 :
        que.push_back(j)

and the last one left in que is the center.

you can prove this by thinking about the diameter path.
to simpify , we assume the length of the diameter path is odd, so that the middle node of the path is unique, let’s call that node M,

we can see that:

1. M will not be pushed to the back of que until every node else on
   diameter path has been pushed into que
2. if there’s another node N
   that is pushed after M has already been pushed into que, then N must
   be on a longer path than the diameter path. Therefore N can’t exist. M must be the last
   node pushed (and left) in que