# PACE 2023 - Input and Output

The task of this year’s challenge is to compute an optimal contraction sequence of a given undirected graph.

## The Input Format

The graph is presented as input in essentially the same graph (.gr) format used in previous iterations of the challenge:

• The vertices of an n-vertex graph are the numbers 1 to n.
• The line separator is \n.
• A line starting with c is a comment and has no semantics.
• The first non-comment line is the p-line:
• It is formatted as p tww n m, where
• n is the number of vertices;
• m the number of edges to follow;
• tww the problem descriptor.
• The p-line is unique and no other line starts with p.
• After the p-line there will be exactly m non-comment lines that encode the edges:
• These are formatted as x y and define an edge between x and y, where x and y are numbers between 1 and n.

Note that this format allows to encode isolated vertices, multi-edges, and self-loops. Here is an example:

c This file describes a graph with 6 vertices and 4 edges.
p tww 6 4
1 2
2 3
c This is a comment and will be ignored.
4 5
4 6


### Contracting two Vertices

The contraction of two vertices $x$ and $y$ is defined as follows:

• any edge (black or red) between $x$ and $y$ gets deleted;
• $x$ retains all black edges to its neighbors that are adjacent to $y$;
• all edges from $x$ to vertices that are not adjacent to $y$ become red;
• $x$ is connected with a red edge to all vertices that are connected to $y$ but not to $x$;
• $y$ gets deleted from the graph.

Note that in this definition, the contraction operation is not symmetric, that is, contracting $(x,y)$ results in a different (but isomorphic) graph than contracting $(y,x)$.

## Contraction Sequences

A contraction sequence for a graph $G=G_1$ is a sequence of vertex pairs $(x_1,y_1),(x_2,y_2),\dots$ such that:

• $x_1$ and $y_1$ are (not necessarily connected) vertices in $G_1$;
• contracting $x_1$ and $y_1$ yields a new graph $G_2$, in which the vertices $x_2$ and $y_2$ are present;
• the same argument holds inductively, that is, $x_i$ and $y_i$ are vertices in $G_i$;
• after all contractions have been performed, a single vertex remains.

### Validity of a Contraction Sequence

A contraction sequence is valid if:

• it contracts vertices at time $i$ that are present at time $i$;
• it produces a single vertex graph.

The width of a contraction sequence is the maximum red degree that appears during the contraction process.

## Output Format

The output of this year’s challenge is the twinwidth format tww that contains:

• Optional comment lines starting with c (as in the input);
• exactly $n-1$ contraction lines formatted as x y.

Here is an example:

1 2
1 3
4 5
4 6
1 6


## Verifier

This Python script (version from 15.02.2023) can be used to check whether a given contraction sequence is valid for a given graph. The output will either be the sequence width or an error message.

 python verifier.py <graph.gr> <result.tww>