Orientable as Desired

AtCoder

IDagc071_c

Time2000ms

Memory256MB

Difficulty—

There is a simple connected undirected graph $G$ with $N$ vertices and $M$ edges, where the vertices are labeled $1,2,\dots,N$. The $i$\-th edge connects two vertices $u_i$ and $v_i$. Determine whether there exists a length-$N$ sequence of non-negative integers $X=(X_1,X_2,\dots,X_N)$ that satisfies all of the following conditions: * $X_v$ does not exceed the degree of vertex $v$ in $G$ for each $v=1,2,\dots,N$. * There is **no** directed graph obtained by assigning directions to the $M$ edges of $G$ in which, for each $v=1,2,\dots,N$, either the in-degree or the out-degree of $v$ equals $X_v$. You have $T$ test cases; solve each of them. ## Constraints * $1 \leq T \leq 10^5$ * $2 \leq N \leq 2 \times 10^5$ * $N-1 \leq M \leq 2 \times 10^5$ * $1 \leq u_i, v_i \leq N$ * All input values are integers. * The graph $G$ given in the input is a simple connected undirected graph. * The sum of $N$ over all test cases in each input is at most $2 \times 10^5$. * The sum of $M$ over all test cases in each input is at most $2 \times 10^5$. ## Input The input is given from Standard Input in the following format: $T$ $\mathrm{case}_1$ $\vdots$ $\mathrm{case}_T$ Each case is given in the following format: $N$ $M$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_M$ $v_M$ [samples]

Samples

Input #1

Output #1

Yes
No
Yes

In the first test case, for example, $X=(2,2,2,1)$ satisfies the conditions.
In the second test case, no sequence $X$ satisfies the conditions.

API Response (JSON)

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      "content": "There is a simple connected undirected graph $G$ with $N$ vertices and $M$ edges, where the vertices are labeled $1,2,\\dots,N$. The $i$\\-th edge connects two vertices $u_i$ and $v_i$. Determine whethe",
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      "content": "There is a simple connected undirected graph $G$ with $N$ vertices and $M$ edges, where the vertices are labeled $1,2,\\dots,N$. The $i$\\-th edge connects two vertices $u_i$ and $v_i$.\nDetermine whethe...",
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