> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/graph-theory/euler.md).

# 欧拉回路

### 概念

* **欧拉路径**：经过每条边恰好一次的路径
* **欧拉回路**：起点和终点重合的欧拉路径

### 存在条件

**无向图**：

* 欧拉回路：所有顶点度数为偶数，且图连通
* 欧拉路径：恰有 0 个或 2 个奇度顶点（2 个时从其中一个出发）

**有向图**：

* 欧拉回路：每个点入度 = 出度，且基图连通
* 欧拉路径：至多一个点出度比入度大 1（起点），至多一个点入度比出度大 1（终点）

### 求法（Hierholzer）

DFS 找回路，回溯时记录边，最后逆序输出。

```cpp
vector<int> ans;
void dfs(int u) {
    while (!adj[u].empty()) {
        int v = adj[u].back(); adj[u].pop_back();
        dfs(v);
        // 无向图需删除反向边
    }
    ans.push_back(u);
}
// 输出时逆序
```

### 应用

一笔画问题、邮递员问题、DNA 拼接等。


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