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D - Unicyclic Components

Score : $400$ points

Problem Statement

You are given an undirected graph with $N$ vertices numbered $1$ to $N$ and $M$ edges numbered $1$ to $M$. Edge $i$ connects vertex $u_i$ and vertex $v_i$.

Determine whether every connected component in this graph satisfies the following condition.

• The connected component has the same number of vertices and edges.

Notes

An undirected graph is a graph with edges without direction.
A subgraph of a graph is a graph formed from a subset of vertices and edges of that graph.
A graph is connected when one can travel between every pair of vertices in the graph via edges.
A connected component is a connected subgraph that is not part of any larger connected subgraph.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $0 \leq M \leq 2 \times 10^5$
• $1 \leq u_i \leq v_i \leq N$
• All values in the input are integers.

Input

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

$N$ $M$

$u_1$ $v_1$

$\vdots$

$u_M$ $v_M$

Output

If every connected component satisfies the condition, print Yes; otherwise, print No.

3 3
2 3
1 1
2 3

Yes

The graph has a connected component formed from just vertex $1$, and another formed from vertices $2$ and $3$.
The former has one edge (edge $2$), and the latter has two edges (edges $1$ and $3$), satisfying the condition.

5 5
1 2
2 3
3 4
3 5
1 5

Yes

13 16
7 9
7 11
3 8
1 13
11 11
6 11
8 13
2 11
3 3
8 12
9 11
1 11
5 13
3 12
6 9
1 10

No