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I saw on Wikipediathat Hierholzer's algorithm has a time complexity of O(E), but won't the dfs used in this algorithm would cause the total time complexity to be O(V+E) instead of only O(E). Assuming we performed the dfs on an Adjacency list. Assume the graph is a connected graph with only one component.

Thanks for the help.

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I'm going to assume that you have a reasonable representation of the graph (for example adjacency lists with vertices indexed by small integers).

The only interesting case is $E \in o(|V|)$ (otherwise you already have $O(|V|+|E|) = O(|E|)$).

Then there are at most $O(|E|)$ non isolated vertices. You can find the set $S$ of these vertices (and hence construct the subgraph induced by $S$) in time $O(|E|)$ by simply creating an array (of size at most $2|E|$) containing all endpoints of the edges in $E$, sorting them (in linear time), and dropping the duplicates (again, in linear time).

Then you can apply the algorithm on the remaining subgraph with has $\Theta(|E|)$ vertices and $|E|$ edges (besides, if the number of remaining vertices is larger than $|E|+1$, then the graph has at least two connected components containing at least one edge each, hence no Eulerian path exists).

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  • $\begingroup$ Are saying that if an Eulerian path exists then E > V hence O(V+E) = O(E) is this assumption right? Lets assume the graph is a connected graph with only one component $\endgroup$ Feb 10, 2022 at 18:54
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    $\begingroup$ No, there are graphs with $|E|=o(|V|)$ that admit an Eulerian path. However, these graphs have isolated vertices. Once you remove then then you are left with $V \le 2|E|$, i.e., $|E| \ge |V|/2$ and hence $O(|V|+|E|)=O(|E|)$. $\endgroup$
    – Steven
    Feb 10, 2022 at 18:58
  • $\begingroup$ I See that makes sense. Thanks $\endgroup$ Feb 10, 2022 at 19:00
  • $\begingroup$ I see that you added "Assume the graph is a connected graph with only one component". In this case you clearly have $|E| \ge |V|-1$ (otherwise the graph won't be connected) and hence $O(|V|+|E|) = O(|E|)$. $\endgroup$
    – Steven
    Feb 10, 2022 at 19:01

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