A small history of domination

The concept of domination, and more generally of graph theory, started in recreation and military problems.

It has been studied by some of the greatest mathematicians and is a lively and hard topic.

We will go over some of these motivating problems

• Constantine the Great (274–337) defense scheme
• Puzzles brainteasers
• Chess problems

A small history of domination

Constantine problem:

How to minimise the legions needed to protect the cities in the Roman empire?

His solution:

Place at most 2 legions in cities such that all cities are neighbouring one legion.

Graph: (Roman domination Cockayne, Dreyer Sr., S.M. Hedetniemi, S.T. Hedetniemi, (2004) Discret. Math. 278)

Colour the graph's verticies with 0,1,2 and make sure all vertices of colour 0 are neighbour of at least one with colour 2.

A small history of domination

Euler (1706–1783)

• bridges of Königsberg;
• knight tour;
• Euler's characteristic: \(V-E+F=2\).

A small history of domination

8 queens problem

Place 8 queens on the chessboard so that they cannot threaten each others. How many solutions are there?

• Max Bezzel introduced it 1848
• Gauss and Schumacher discussed it 1850
• Nauck published the 92 solutions in 1850
• known up to \(n=27\) (2016).

A small history of domination

5 queens problem

Place 5 queens on the chessboard to threaten all squares.

• De Jaenisch introduced it after a letter in 1863
• Sparked a lot of interest
• known up to \(n=32\). (2022)
• unknown if the number of queens needed always increases (!)

Conjecture (Hedetiemi ~1992)

The domination number grows monotonously with \(n\).

Chess to graph theory

Primer on graph theory

Graph: \( (V,E) \), set of vertices \(V\) and set of edges \(E\).

Bipartite: The vertices divide in two sets with edges only between the two sets.

Planar: the graph can be placed with edges that do not cross.

Example:

Chess to graph theory

Primer on graph theory  Graph \(G=(V,E)\)

Clique: A subset of vertices all connected to each others

Coclique: A subset of vertices with no edge between them

Domination: find a subset of vertices to which all vertices are attached.

Example:

Chess to graph theory

Chess graph

Choose a piece. The vertices are the square of the chessboard, and there is an edge between two vertices if the piece can travel between the two.

Example:

Chess to graph theory

Converting problems into graph

We will study those problems for queen and rook graphs on slightly different chessboards.

Chess to graph theory

Some notation

MinDom(I)R: minimal (independent) domination on rook graph.

MinDom(I)Q: minimal (independent) domination on queen graph.

MaxR: maximal coclique on rook graph.

MaxQ: maximal coclique on queen graphs.

Chess to graph theory

Some results on chessboard of size \(n \)

MinDom(I)R: trivial (\(n\) rooks);

MinDom(I)Q: unknown;

MaxR: trivial (\(n\) rooks);

MaxQ: easy (\(n\) queens after \(n\geq 4\), an explicit solution for each \(n\) was given by Emil Pauls in 1873)

Polycubes

Polyomino

1) A polyomino is a connected set of edge-connected tiles.

2) A rook moving on an infinite chessboard defines a polyomino.

Polycubes

The generalisation to dimension \(d\).

1) a set of face-connected hypercubes.

Example:

Polycubes

Queen and rook movement

A rook moves through faces. A queen moves through all intersections (in 3D: faces, edges and vertices).

Example: link rook                                                                       link queen

Polycubes

Domination on polycubes

Define the graph on polycubes. Only concern: you cannot jump.

Polycubes

Problems

We will study the MinDomR/Q and MaxR/Q problems on polycubes.

One change:

We cannot say how many will be necessary knowing only the number of tiles of the polycube

Polycubes

Proposition (Alpert–Roldán-Roa)

1. Given a polycube of size \(n\geq 2\), the minimal number of rooks needed lies between 1 and \(\lfloor \frac{n}{2}\rfloor\).
2. Given a polycube of size \(n\geq 3\), the minimal number of queens needed lies between 1 and \(\lfloor \frac{n}{3}\rfloor\).

Polycubes

Proposition (Alpert–Roldán-Roa)

1. Given a polycube of size \(n\geq 2\), the minimal rook domination number lies between 1 and \(\lfloor \frac{n}{2}\rfloor\).
2. Given a polycube of size \(n\geq 3\), the minimal queen domination number lies between 1 and \(\lfloor \frac{n}{3}\rfloor\).

Proof 1:

Polycubes

Proposition (Alpert–Roldán-Roa)

1. Given a polycube of size \(n\geq 2\), the minimal rook domination number lies between 1 and \(\lfloor \frac{n}{2}\rfloor\).
2. Given a polycube of size \(n\geq 3\), the minimal queen domination number lies between 1 and \(\lfloor \frac{n}{3}\rfloor\).

Proof 2:

COmputational complexity

We will quantify how hard the problem is instead.

P

There is a polynomial algorithm.

NP-Complete

1. We can verify a solution in polynomial time.
2. It is equivalent to SAT
3. As difficult as any other NP-complete problem.
4. No known polynomial algorithm exists.

COmputational complexity

Proposition (A–RR 2021 and LR–M–RR 2022)

MinDomR and MinDomQ are NP-complete.

So it's hard. Let's test how!

https://phoenixsmaug.itch.io/polyomino

COmputational complexity

Proposition (A-RR 2021)

MaxR is in P on polyominoes

Proof:

COmputational complexity

Question (A-RR 2021)

What is the complexity of MaxR/Q on polycubes of dimension \(d\geq 3\)?

Our results

Proposition (LR-M-RR 2022)

MaxR and MaxQ are NP-complete on polycubes of size \(d\geq 3\).

Proof:

1. Prove that verifying a solution is polynomial
2. Reduce the problem to a known NP-complete problem in polynomial time.

Our results

Proposition (LR-M-RR 2022)

MaxR/Q are NP-complete on polycubes of size \(d\geq 3\).

Proof:

1. Prove that verifying a solution is polynomial

Easy. Given a proposed placement, we check that everything is covered and it grows only polynomially with the size of the board.

Our results

Proposition (LR-M-RR 2022)

MaxR and MaxQ are NP-complete on polycubes of size \(d\geq 3\).

Proof:

2) Reduce to a known NP-complete problem

This is harder. We go for P3SAT3

Our results

P3SAT3

• Set of Boolean variable
• Set of clauses with 2 or 3 literals
• Bipartite clause graph is planar
• Bipartite clause graph has exactly three edges per literals

Our results

We will go from an instance of P3SAT3 to a rook domination problem on polycubes.

We need:

1. Literal gadgets
2. Splitting gadget
3. Clause evalutation gadgets

Literal gadgets

Model

Splitting gadgets

Model

Clause gadgets

\(x_1\vee x_2\) or \(\bar x_1\vee \bar x_2\)

Model

Clause gadgets

\(x_1\vee \bar x_2\)

Model

Clause gadgets

\(x_1\vee x_2\vee x_3\) or \(\bar x_1\vee \bar x_2\vee \bar x_3\)

Model

Clause gadgets

\(\bar x_1\vee x_2\vee  x_3\) or \(x_1\vee \bar x_2\vee  \bar x_3\)

Model

Putting clauses

\( x_1\vee x_2\vee  x_3\)

Model

Translation

1. \(X\) an instance of P3SAT3

2. Construct polycube \(P(X)\)

3. From the gadgets, we know its guarding number \(N\) will respect:

4. The instance is true if and only \( N= M + x_{Clause}\). (All clauses are true.)

Translation

Sanity check: is the polycube too big?

Thank you!