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### Graphs represented by words

Joint work with

Magnus M. Halldorsson

Sergey Kitaev

Reykjavik University

Reykjavik University

Artem Pyatkin

Sobolev Institute of Mathematics

Basic definitions

A finite word over {x,y} is alternating if it does not contain xx and yy.

Alternating words: yx, xy, xyxyxyxy, yxy, etc.

Non-alternating words: yyx, xyy, yxxyxyxx, etc.

Letters x and yalternate in a word w if they induce an alternating subword.

x and y alternate in w = xyzazxayxzyax

Graphs represented by words

Basic definitions

A finite word over {x,y} is alternating if it does not contain xx and yy.

Alternating words: yx, xy, xyxyxyxy, yxy, etc.

Non-alternating words: yyx, xyy, yxxyxyxx, etc.

Letters x and yalternate in a word w if they induce an alternating subword.

x and y alternate in w = xyzazxayxzyax

x and y do not alternate in w = xyzazyaxyxzyax

Graphs represented by words

Basic definitions

A word w is k-uniform if each of its letters appears in w exactly k times.

A 1-uniform word is also called a permutation.

word-representant

- A graph G=(V,E) is represented by a word w if
- Var(w)=V, and
- (x,y) Eiffx and y alternate in w.

A graph is (k-)representable if it can be represented by a (k-uniform) word.

A graph G is 1-representableiff G is a complete graph.

Graphs represented by words

Example of a representable graph

y

Switching the indicated x and

a would create an extra edge

v

x

cycle graph

z

a

xyzxazvay represents the graph

xyzxazvayv 2-represents the graph

Graphs represented by words

Cliques and Independent Sets

V={A,B,C,...Z}

W=ABC...Z ABC...Z

Kn

Kn

Clique

Independent set

W=ABC...YZ ZY...CBA

Graphs represented by words

Original motivation to study such representable graphs: The Perkins semigroup

S. Kitaev, S. Seif: Word problem of the Perkins semigroup via directed

acyclic graphs, Order (2009).

Related work: Split-pair arrangement (application: scheduling robots on a path, periodically)

R. Graham, N. Zhang: Enumerating split-pair arrangements,

J. Combin. Theory A, Feb. 2008.

Graphs represented by words

Papers on representable graphs:

S. Kitaev, A. Pyatkin: On representable graphs, Automata, Languages

and Combinatorics (2008).

M. Halldorsson, S. Kitaev, A. Pyatkin: On representable graphs,

semi-transitive orientations, and the representation numbers, preprint.

Graphs represented by words

Operations Preserving Representability

- Replacing a node v by a module H
- H can be any clique or any comparability graph
- Neighbors of v become neighbors of all nodes in H
- Gluing two representable graphs at 1 node
- Joining two representable graphs by an edge

G

+

H

=

G

H

G

&

H

=

G

H

Graphs represented by words

Operations Not Preserving Representability

- Taking the line graph
- Taking the complement
- Attaching two graphs at more than 1 node

Open question: Does it preserve non-representability?

G

+

H

=

G

H

The graph in red is not 2- or 3-representable. It is not known if it is representable or not.

Graphs represented by words

Properties of representable graphs

If G is k-representable and m>k then G is m-representable.

If G is representable then G is k-representable for some k.

For representable graphs, we may restrict ourselves to connected graphs.

G U H (G and H are two connected components) is representable iff

G and H are representable. (Take concatenation of the corresponding

words representants having at least two copies of each letter.)

Graphs represented by words

2-representable graphs

- 1-representable graphs cliques
- 2-representable graphs ??
- A B C D E F G H C D H G F A B D

Graphs represented by words

2-representable graphs

- View as overlapping intervals: u & v adjacent if they overlap

Example:

A B C D E F G H C D H G FA B E

uv E

u

v

E

A

F

Graphs represented by words

2-representable graphs

- View as overlapping intervals:

Equivalent to Interval overlap graphs

A B C D E F G H C D H G FA B E

E

A

F

Graphs represented by words

2-representable graphs

A B C D E F G H C D H G F A B E

A B C D E F G H C D H G F A B E

Graphs represented by words

Comparability graphs

- We can orient the edges to form a transitive digraph
- They correspond to partial orders.

Graphs represented by words

Comparability graphs

- We can orient the edges to form a transitive digraph
- They correspond to partial orders.

Graphs represented by words

Comparability graphs

- We can orient the edges to form a transitive digraph
- They correspond to partial orders.

Graphs represented by words

Representing comparability graphs

- Form a topological ordering, where a given letter, say c, is as early as possible:abcdefg

a

b

c

d

f

e

g

Graphs represented by words

Representing comparability graphs

- Form a topological ordering, where a given letter, say c, is as early as possible:abcdefg
- Then add another where it is as late as possible abfgdce
- Repeat from 1. until done

a

b

c

d

f

e

g

Graphs represented by words

Representing comparability graphs

- The resulting substringabcdefg abfgdcecoversall non-edges incident on c.

a

b

c

d

f

e

g

Graphs represented by words

Representing comparability graphs

- The resulting substringabcdefg abfgdcecoversall non-edges incident on c.
- For this graph it would suffice to repeat this for f:abfgcde abcdefgplus one round for d: dabcdfg
- Final string:

a

b

c

d

f

e

g

abcdefg abfgdce abfgcde abcdefg dabcdfg

Graphs represented by words

Properties of representable graphs

A graph is permutationally representable if it can be

represented by a word of the form P1P2...Pk where

Pis are permutations of the same set.

2

4

is permutationally representable (13243142)

3

1

Lemma (Kitaev and Seif). A graph is permutationally representable iff it

is transitively orientable, i.e. if it is a comparability graph.

Graphs represented by words

Shortcut – a type of digraph

- Acyclic, non-transitive
- Contains directed cycle a, b, c, d, except last edge is reversed
- Non-transitive Not representable

a

Missing!

b

c

d

Graphs represented by words

Main result

- A graph G is representable iff G is orientable to a shortcut-free digraph
- () Straightforward.
- () We give an algorithm that takes any shortcut-free digraph and produces a word that represents the graph

Graphs represented by words

Sketch of our algorithm

- Chain together copies of the digraph (= D’)
- If ab D, then biai+1 D’

a

b

c

d

Graphs represented by words

Sketch of our algorithm

- Chain together copies of the digraph (= D’)
- If ab D, then biai+1 D’

a

b

c

d

a

b

c

d

Graphs represented by words

Sketch of our algorithm

We allow the topsort to traverse the 2nd copy before finishing the 1st . The added edges ensure that adjacent nodes still alternate.

- Chain together copies of the digraph (= D’)
- If ab D, then biai+1 D’
- Form a topsort of D’ of pairs of copies.
- In 1st copy, some letter d occurs as late as possible
- In 2nd copy d occurs as early as possible

a

b

c

d

a

b

c

Example:

a b c

a

d

cbd

d

Graphs represented by words

Size of the representation

- The algorithm creates a word where each of the n letters appears at most n times.

Each representable graph is n-representable

- There are graphs that require n/2 occurrences
- E.g. based on the cocktail party graph
- Deciding whether a given graph is k-representable, for k between 3 and [n/2], is NP-complete

Graphs represented by words

Corollary: 3-colorable graphs

- 3-colorable graphs are representable
- Red->Green->Blue orientation is shortcut-free!

Graphs represented by words

Non-representable graphs

Lemma. Let x be a vertex of degree n-1 in G having n nodes.

Let H=G \ {x}. Then G is representable iff H is permutationally

representable.

The lemmas give us a method to construct non-representable graphs.

Graphs represented by words

Construction of non-representable graphs

- Take a graph that is not a comparability graph (C5 is
- the smallest example);
- Add a vertex adjacent to every node of the graph;
- Add other vertices and edges incident to them (optional).

W5 – the smallest non-representable graph

All odd wheelsW2t+1 for t ≥ 2

are non-representable graphs.

Graphs represented by words

Small non-representable graphs

Graphs represented by words

Relationships of graph classes

4-colorable & K4-free

Representable

Perfect

Circle

2-repres.

3-colorable

Comparability

Planar

Chordal

2-inductive

2-outerplanar

Bipartite

3-trees

Partial 2-trees

Split

Outerplanar

2-trees

Trees

Graphs represented by words

A property of representable graphs

G[N(x)] = graph induced by the neighborhood of x

G representable For each x V, G[N(x)] is permutationally representable,

Main means of showing non-representability

Natural question: Is the converse statement true?

Graphs represented by words

A non-representable graph whose induced neighborhood graphs are all comparability

co-T2

T2

Graphs represented by words

3-representable graphs

examples of prisms

Theorem (Kitaev, Pyatkin). Every prism is 3-representable.

Theorem (Kitaev, Pyatkin). For every graph G there exists a 3-representable graph H that contains G as a minor.

In particular, a 3-subdivision of every graph G is 3-representable.

Graphs represented by words

One more result

We can construct graphs with represntation number k=[n/2]

Coctail party graph:

Graphs represented by words

One more result

We can construct graphs with represntation number k=[n/2]

Coctail party graph:

Graphs represented by words

Complexity

- Recognizing representable graphs is in NP
- Certificate is an orientation
- Is it NP-hard?
- Most optimization problems are hard
- Ind. Set, Dom. Set, Coloring, Clique Partition...
- Max Clique is polynomially solvable on repr.gr.
- A clique is contained within some neighborhood
- Neighborhoods induce comparability graphs

Open!

Graphs represented by words

Open problems

Open!

- Is it NP-hard to decide whether a given graph is representable?
- What is the maximum representation number of a graph (between n/2 and n)?
- Can we characterize the forbidden subgraphs of representative graphs?
- Graphs of maximum degree 4?
- How many (k-)representable graphs are there?

Graphs represented by words

Resolved question

1

Is the Petersen’s graph

representable?

6

Yes!

2

5

10

7

9

8

It is 3-representable:

3

4

1,3,8,7,2,9,6,10,7,4,9,3,5,4,1,2,8,3,10,7,6,8,5,10,1,9,4,5,6,2

Graphs represented by words

Resolved questions

Are there any non-representable graphs with N(v) inducing a comparability graphs for every vertex v? In particular,

- Are there non-representable graphs of maximum degree 3?
- Are there 3-chromatic non-representable graphs?
- Are there any triangle-free non-representable graphs?

Yes!

No!

No!

Yes!

Graphs represented by words

Open/Resolved problems

- Is it NP-hard to determine whether a given graph is NP-representable.
- Is it true that every representable graph is k-representable for some k?
- How many (k-)representable graphs on n vertices are there?

Open!

Yes!

Open!

Graphs represented by words

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