Everywhere is Sparser than Whole (Judge)

AtCoder

IDarc161_f

Time8000ms

Memory256MB

Difficulty—

We define the **density** of a non-empty simple undirected graph as $\displaystyle\frac{(\text{number\ of\ edges})}{(\text{number\ of\ vertices})}$. You are given positive integers $N$, $D$, and a simple undirected graph $G$ with $N$ vertices and $DN$ edges. The vertices of $G$ are numbered from $1$ to $N$, and the $i$\-th edge connects vertex $A_i$ and vertex $B_i$. Determine whether $G$ satisfies the following condition. **Condition:** Let $V$ be the vertex set of $G$. For any non-empty **proper** subset $X$ of $V$, the density of the induced subgraph of $G$ by $X$ is **strictly less than** $D$. There are $T$ test cases to solve. What is an induced subgraph? For a vertex subset $X$ of graph $G$, the **induced subgraph** of $G$ by $X$ is the graph with vertex set $X$ and edge set containing all edges of $G$ that connect two vertices in $X$. In the above condition, note that we only consider vertex subsets that are neither empty nor the entire set. ## Constraints * $T \geq 1$ * $N \geq 1$ * $D \geq 1$ * The sum of $DN$ over the test cases in each input file is at most $5 \times 10^4$. * $1 \leq A_i < B_i \leq N \ \ (1 \leq i \leq DN)$ * $(A_i, B_i) \neq (A_j, B_j) \ \ (1 \leq i < j \leq DN)$ ## Input The input is given from Standard Input in the following format: $T$ $\mathrm{case}_1$ $\mathrm{case}_2$ $\vdots$ $\mathrm{case}_T$ Each test case $\mathrm{case}_i \ (1 \leq i \leq T)$ is in the following format: $N$ $D$ $A_1$ $B_1$ $A_2$ $B_2$ $\vdots$ $A_{DN}$ $B_{DN}$ [samples]

Samples

Input #1

Output #1

Yes
No

*   The first test case is the same as Sample Input 1 in [Problem D](./arc161_d), and it satisfies the condition.
*   For the second test case, the edge set of the induced subgraph by the non-empty proper subset ${1, 2, 3}$ of the vertex set ${1, 2, 3, 4}$ is ${(1, 2), (1, 3), (2, 3)}$, and its density is $\displaystyle\frac{3}{3} = 1 = D$. Therefore, this graph does not satisfy the condition.

API Response (JSON)

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