Problem:
Score : 800 points

Problem Statement

Takahashi has received an undirected graph with N vertices, numbered 1, 2, ..., N.
The edges in this graph are represented by (u_i, v_i).
There are no self-loops and multiple edges in this graph.

Based on this graph, Takahashi is now constructing a new graph with N^2 vertices, where each vertex is labeled with a pair of integers (a, b) (1 \leq a \leq N, 1 \leq b \leq N).
The edges in this new graph are generated by the following rule:

Span an edge between vertices (a, b) and (a', b') if and only if both of the following two edges exist in the original graph: an edge between vertices a and a', and an edge between vertices b and b'.

How many connected components are there in this new graph?

Constraints

2 \leq N \leq 100,000

0 \leq M \leq 200,000

1 \leq u_i < v_i \leq N

There exists no pair of distinct integers i and j such that u_i = u_j and v_i = v_j.

Input

The input is given from Standard Input in the following format:

N M
u_1 v_1
u_2 v_2
:
u_M v_M

Output

Print the number of the connected components in the graph constructed by Takahashi.

Sample Input 1

3 1
1 2

Sample Output 1

7

The graph constructed by Takahashi is as follows.

Sample Input 2

7 5
1 2
3 4
3 5
4 5
2 6

Sample Output 2

18

Solution in Rust (1.15.1) (judge verdict: Accepted)

#![allow(unused_imports)]
#![allow(non_snake_case)]
use std::cmp::*;
use std::collections::*;
use std::io::Write;

#[allow(unused_macros)]
macro_rules! debug {
    ($($e:expr),*) => {
        #[cfg(debug_assertions)]
        $({
            let (e, mut err) = (stringify!($e), std::io::stderr());
            writeln!(err, "{} = {:?}", e, $e).unwrap()
        })*
    };
}

fn main() {
    let v = read_vec::<usize>();
    let (n, m) = (v[0], v[1]);
    let mut pairs = vec![];
    for _ in 0..m {
        let v = read_vec::<usize>();
        let (a, b) = (v[0] - 1, v[1] - 1);
        pairs.push((a, b));
    }
    let mut uft = UnionFindTree::new(n);
    for &(a, b) in pairs.iter() {
        uft.unite(a, b);
    }
    let mut used_root = HashSet::new();
    for i in 0..n {
        used_root.insert(uft.find(i));
    }
    let mut solver = Solver::new(n);
    for &(a, b) in pairs.iter() {
        solver.adj_mat[a].push(b);
        solver.adj_mat[b].push(a);
    }

    let mut isolated = 0;
    let mut biparate = 0;
    let mut etc = 0;
    for root in used_root {
        if uft.get_size(root) == 1 {
            isolated += 1;
        } else if solver.solve(root, 1) {
            biparate += 1;
        } else {
            etc += 1;
        }
    }
    // debug!(isolated);
    let ans = (isolated * isolated)
        + 2 * isolated * (n - isolated)
        + 2 * biparate * biparate
        + etc * etc
        + 2 * biparate * etc;
    println!("{}", ans);
}

fn read<T: std::str::FromStr>() -> T {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).ok();
    s.trim().parse().ok().unwrap()
}

fn read_vec<T: std::str::FromStr>() -> Vec<T> {
    read::<String>()
        .split_whitespace()
        .map(|e| e.parse().ok().unwrap())
        .collect()
}

#[derive(Debug, Clone)]
struct UnionFindTree {
    parent: Vec<isize>,
    size: Vec<usize>,
    height: Vec<u64>,
}

impl UnionFindTree {
    fn new(n: usize) -> UnionFindTree {
        UnionFindTree {
            parent: vec![-1; n],
            size: vec![1usize; n],
            height: vec![0u64; n],
        }
    }

    fn find(&mut self, index: usize) -> usize {
        if self.parent[index] == -1 {
            return index;
        }
        let idx = self.parent[index] as usize;
        let ret = self.find(idx);
        self.parent[index] = ret as isize;
        ret
    }

    fn same(&mut self, x: usize, y: usize) -> bool {
        self.find(x) == self.find(y)
    }

    fn get_size(&mut self, x: usize) -> usize {
        let idx = self.find(x);
        self.size[idx]
    }

    fn unite(&mut self, index0: usize, index1: usize) -> bool {
        let a = self.find(index0);
        let b = self.find(index1);
        if a == b {
            false
        } else {
            if self.height[a] > self.height[b] {
                self.parent[b] = a as isize;
                self.size[a] += self.size[b];
            } else if self.height[a] < self.height[b] {
                self.parent[a] = b as isize;
                self.size[b] += self.size[a];
            } else {
                self.parent[b] = a as isize;
                self.size[a] += self.size[b];
                self.height[a] += 1;
            }
            true
        }
    }
}

struct Solver {
    adj_mat: Vec<Vec<usize>>,
    colors: Vec<i32>, // 0 uncolored, 1 or -1
}

impl Solver {
    fn new(n: usize) -> Solver {
        Solver {
            adj_mat: vec![vec![]; n],
            colors: vec![0; n],
        }
    }
    fn solve(&mut self, idx: usize, color: i32) -> bool {
        if self.colors[idx] != 0 {
            return self.colors[idx] == color;
        }
        self.colors[idx] = color;
        for i in 0..self.adj_mat[idx].len() {
            let to = self.adj_mat[idx][i];
            if self.colors[to] == 0 {
                let result = self.solve(to, color * -1);
                if !result {
                    return false;
                }
            } else if self.colors[to] == color {
                return false;
            }
        }
        true
    }
}