Lights Out on Connected Graph

AtCoder

IDarc105_f

Time3000ms

Memory256MB

Difficulty—

Given is an undirected graph $G$ consisting of $N$ vertices numbered $1$ through $N$ and $M$ edges numbered $1$ through $M$. It is guaranteed that $G$ is connected and contains no self-loops and no multi-edges. Edge $i$ connects Vertex $a_i$ and Vertex $b_i$ bidirectionally. Each edge can be painted red or blue. Initially, every edge is painted red. Consider making a new graph $G^{\prime}$ by removing zero or more edges from $G$. There are $2^M$ graphs that $G^{\prime}$ can be. Among them, find the number of good graphs defined below, modulo $998244353$. $G^{\prime}$ is said to be a _good graph_ when both of the following conditions are satisfied: * $G^{\prime}$ is connected. * It is possible to make all edges blue by doing the following operation zero or more times. * Choose one vertex and change the color of every edge incident to that vertex, that is, red to blue and vice versa. ## Constraints * All values in input are integers. * $1 \leq N \leq 17$ * $N-1 \leq M \leq N(N-1)/2$ * $G$ is connected and has no self-loops and no multi-edges. ## Input Input is given from Standard Input in the following format: $N$ $M$ $a_1$ $b_1$ $\vdots$ $a_M$ $b_M$ [samples]

Samples

Input #1

3 2
1 2
2 3

Output #1

1

*   The conditions are satisfied only if neither Edge $1$ nor Edge $2$ is removed.
    *   In this case, we can make all edges blue by, for example, doing the operation for Vertex $2$.
*   Otherwise, the graph gets disconnected and thus does not satisfy the condition.

Input #2

Output #2

Input #3

Output #3

90625632

*   Be sure to find the count modulo $998244353$.

API Response (JSON)

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  "problem": {
    "name": "Lights Out on Connected Graph",
    "description": {
      "content": "Given is an undirected graph $G$ consisting of $N$ vertices numbered $1$ through $N$ and $M$ edges numbered $1$ through $M$. It is guaranteed that $G$ is connected and contains no self-loops and no mu",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc105_f"
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  "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 $M$. It is guaranteed that $G$ is connected and contains no self-loops and no mu...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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