CF1364D.Ehab's Last Corollary

Given a connected undirected graph with $n$ vertices and an integer $k$, you have to either:

• either find an independent set that has exactly $\lceil\frac{k}{2}\rceil$ vertices.
• or find a simple cycle of length at most $k$.

An independent set is a set of vertices such that no two of them are connected by an edge. A simple cycle is a cycle that doesn't contain any vertex twice.

I have a proof that for any input you can always solve at least one of these problems, but it's left as an exercise for the reader.

给定一个包含 $n$ 个顶点的连通无向图以及一个整数 $k$，你需要完成以下二者之一：

• 找到一个恰好包含 $\lceil\frac{k}{2}\rceil$ 个顶点的独立集；
• 或找到一个长度至多为 $k$ 的简单环。

独立集是指其中任意两个顶点之间均无边相连的顶点集合；简单环是指不重复经过任何顶点的环。

我有一个证明，表明对任意输入，上述两个问题中至少有一个总能被解决，但该证明留作读者练习。

The first line contains three integers $n$, $m$, and $k$ ($3 \le k \le n \le 10^5$, $n-1 \le m \le 2 \cdot 10^5$) — the number of vertices and edges in the graph, and the parameter $k$ from the statement.

Each of the next $m$ lines contains two integers $u$ and $v$ ($1 \le u,v \le n$) that mean there's an edge between vertices $u$ and $v$. It's guaranteed that the graph is connected and doesn't contain any self-loops or multiple edges.

第一行包含三个整数 $n$、$m$ 和 $k$（$3 \le k \le n \le 10^5$，$n-1 \le m \le 2 \cdot 10^5$），分别表示图中的顶点数、边数以及题目中所述的参数 $k$。

接下来的 $m$ 行中，每行包含两个整数 $u$ 和 $v$（$1 \le u,v \le n$），表示顶点 $u$ 与顶点 $v$ 之间存在一条边。保证该图是连通的，且不包含自环或重边。

If you choose to solve the first problem, then on the first line print $1$, followed by a line containing $\lceil\frac{k}{2}\rceil$ distinct integers not exceeding $n$, the vertices in the desired independent set.

If you, however, choose to solve the second problem, then on the first line print $2$, followed by a line containing one integer, $c$, representing the length of the found cycle, followed by a line containing $c$ distinct integers not exceeding $n$, the vertices in the desired cycle, in the order they appear in the cycle.

如果你选择解决第一个问题，则在第一行输出 $1$，然后在下一行输出 $\lceil\frac{k}{2}\rceil$ 个互不相同的整数（均不超过 $n$），即所求独立集中的顶点。

如果你选择解决第二个问题，则在第一行输出 $2$，然后在下一行输出一个整数 $c$，表示所找到环的长度，再在下一行输出 $c$ 个互不相同的整数（均不超过 $n$），即所求环中的顶点，按其在环中出现的顺序排列。

In the first sample:

Notice that printing the independent set ${2,4}$ is also OK, but printing the cycle $1-2-3-4$ isn't, because its length must be at most $3$.

In the second sample:

Notice that printing the independent set ${1,3}$ or printing the cycle $2-1-4$ is also OK.

In the third sample:

In the fourth sample:

在第一个样例中：

注意：输出独立集 $\{2,4\}$ 也是可以的，但输出环 $1\text{-}2\text{-}3\text{-}4$ 则不行，因为其长度至多为 $3$。

在第二个样例中：

注意：输出独立集 $\{1,3\}$ 或输出环 $2\text{-}1\text{-}4$ 也是可以的。

在第三个样例中：

在第四个样例中：