USACO 2025 February Contest, Gold
Problem 3. Friendship Editing
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Farmer John's $N$ cows are labeled $1$ to $N$ ($2\le N\le 16$). The friendship relationships between the cows can be modeled as an undirected graph with $M$ ($0\le M\le N(N-1)/2$) edges. Two cows are friends if and only if there is an edge between them in the graph.
In one operation, you can add or remove a single edge from the graph. Count the minimum number of operations required to ensure that the following property holds: If cows $a$ and $b$ are friends, then for every other cow $c$, at least one of $a$ and $b$ is friends with $c$.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains $N$ and $M$.The next $M$ lines each contain a pair of friends $a$ and $b$ ($1\le a<b\le N$). No pair of friends appears more than once.
OUTPUT FORMAT (print output to the terminal / stdout):
The number of edges you need to add or remove.SAMPLE INPUT:
3 1 1 2
SAMPLE OUTPUT:
1
The network violates the property. We can add one of edges $(2,3)$ or $(1,3)$, or remove edge $(1,2)$ to fix this.
SAMPLE INPUT:
3 2 1 2 2 3
SAMPLE OUTPUT:
0
No changes are necessary.
SAMPLE INPUT:
4 4 1 2 1 3 1 4 2 3
SAMPLE OUTPUT:
1
SCORING:
- Inputs 4-13: One input for each $N\in [6, 15]$ in increasing order.
- Inputs 14-18: $N=16$
Problem credits: Benjamin Qi