Suppose $G$ is a DAG with $n$ vertices, and $v$ is a vertex of $G$. What can we say regarding $v$ if the following holds:

A. In all topological sorts, $v$ is at the end of the list.

So my initial answer is that $v$ is the "sink" vertex, but in some case its not always correct, can't seem to find a feature that cant be said on $v$?

B. In all topological sorts, $v$ is at the top of the list.

So this one is the opposite which is the source vertex and has arrows coming out of him and no one points to him.

  • 1
    $\begingroup$ 'the "sink" vertex' -- this phrasing implies a DAG has exactly 1 sink vertex. $\endgroup$ – j_random_hacker May 2 '19 at 15:44
  • $\begingroup$ Is topological cue the result of topological sorting on a DAG? Is your graph a DAG? $\endgroup$ – George Vidalakis May 2 '19 at 15:45
  • $\begingroup$ @GeorgeVidalakis yes and yes $\endgroup$ – G95 May 2 '19 at 15:46
  • $\begingroup$ @j_random_hacker topoligical sorting can bring diffrent results, what can I say for on v of question A? $\endgroup$ – G95 May 2 '19 at 15:49
  • $\begingroup$ @GeorgeVidalakis: A sink is just a vertex with no out-edge. Your example DAG has two sinks. $\endgroup$ – j_random_hacker May 2 '19 at 16:23

A. Let's prove that $v$ will be a vertex which can be reached by every node (it is the mother vertex of the transpose of the graph) by contradiction. Suppose that there is a node $u$ such that $v$ is unreachable by it. Consider every node in the DAG reachable by $u$. There must be at least one node in this set that has no leaving edges. (Otherwise you could start from $u$ and always follow a leaving edge from the current node so after at most $|V| - 1$ steps you would visit an already visited node. But this implies a cycle which isn't allowd in a DAG.) So no other node is reachable by $u$ and $u$ could be last in a topological cue instead of $v$ and we reached a contradiction. As a result $v$ is reachable by every node. This is equivalent with (A)'s statement as a node being reachable by every node can appear only last in any topological cue.

Also $v$ is a sink because if it wasn't there would be at least one edge leaving it so it couldn't be at the end of any topological cue. (A)'s statement implies that $v$ is a sink but the reverse doesn't always hold.

B. $v$ is a mother vertex (every node is reachable by it). The proof is similar.


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