Keep Graph Disconnected

AtCoder

IDarc105_e

Time2000ms

Memory256MB

Difficulty—

Given is an undirected graph $G$ consisting of $N$ vertices numbered $1$ through $N$ and $M$ edges numbered $1$ through $M$. Edge $i$ connects Vertex $a_i$ and Vertex $b_i$ bidirectionally. $G$ is said to be a _good graph_ when both of the conditions below are satisfied. It is guaranteed that $G$ is initially a good graph. * Vertex $1$ and Vertex $N$ are not connected. * There are no self-loops and no multi-edges. Taro the first and Jiro the second will play a game against each other. They will alternately take turns, with Taro the first going first. In each player's turn, the player can do the following operation: * Operation: Choose vertices $u$ and $v$, then add to $G$ an edge connecting $u$ and $v$ bidirectionally. The player whose addition of an edge results in $G$ being no longer a good graph loses. Determine the winner of the game when the two players play optimally. You are given $T$ test cases. Solve each of them. ## Constraints * All values in input are integers. * $1 \leq T \leq 10^5$ * $2 \leq N \leq 10^{5}$ * $0 \leq M \leq \min(N(N-1)/2,10^{5})$ * $1 \leq a_i,b_i \leq N$ * The given graph is a good graph. * In one input file, the sum of $N$ and that of $M$ do not exceed $2 \times 10^5$. ## Input Input is given from Standard Input in the following format: $T$ $\mathrm{case}_1$ $\vdots$ $\mathrm{case}_T$ Each case is in the following format: $N$ $M$ $a_1$ $b_1$ $\vdots$ $a_M$ $b_M$ [samples]

Samples

Input #1

Output #1

First
Second
First

*   In test case $1$, Taro the first wins. Below is one sequence of moves that results in Taro's win:
    *   In Taro the first's turn, he adds an edge connecting Vertex $1$ and $2$, after which the graph is still good.
    *   Then, whichever two vertices Jiro the second would choose to connect with an edge, the graph would no longer be good.
    *   Thus, Taro wins.

API Response (JSON)

{
  "problem": {
    "name": "Keep Graph Disconnected",
    "description": {
      "content": "Given is an undirected graph $G$ consisting of $N$ vertices numbered $1$ through $N$ and $M$ edges numbered $1$ through $M$. Edge $i$ connects Vertex $a_i$ and Vertex $b_i$ bidirectionally. $G$ is sai",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc105_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given is an undirected graph $G$ consisting of $N$ vertices numbered $1$ through $N$ and $M$ edges numbered $1$ through $M$. Edge $i$ connects Vertex $a_i$ and Vertex $b_i$ bidirectionally.\n$G$ is sai...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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