Graphs represented by words

Graphs represented by words Sergey Kitaev

Reykjavik University

Sobolev Institute of Mathematics

Joint work with

Artem Pyatkin

Magnus M. Halldorsson

Reykjavik University

Basic definitions

Sergey Kitaev Graphs represented by words

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 y alternate in a word w if they induce an alternating subword.

x and y alternate in w = xyzazxayxzyax

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 y alternate 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

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.

A graph G=(V,E) is represented by a word w if 1. Var(w)=V, and2. (x,y) E iff x and y alternate in w.

word-representant

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

A graph G is 1-representable iff G is a complete graph.

Example of a representable graph


cycle graph

x

y

v

z a

xyzxazvay represents the graph

xyzxazvayv 2-represents the graph

Switching the indicated x and a would create an extra edge


Cliques and Independent Sets

Kn

Clique

Kn

Independent set

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

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


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.


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.

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+ = H G

G H& = G H

Operations Not Preserving Representability

• Taking the line graph

• Taking the complement

• Attaching two graphs at more than 1 node


G H+ = H G

Open question: Does it preserve non-representability?

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

Properties of representable graphs


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

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

G U H (G and H are two connected components) is representable iffG and H are representable. (Take concatenation of the corresponding words representants having at least two copies of each letter.)

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

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



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

Example:

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


E

A

F

u vuv E


• View as overlapping intervals:

Equivalent to Interval overlap graphs

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


E

A

F



Circle graphs

Comparability graphs

• We can orient the edges to form a transitive digraph

• They correspond to partial orders.


Representing comparability graphs

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


e

b

c

a

g

fd


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

2. Then add another where it is as late as possible abfgdce

3. Repeat from 1. until done


e

b

c

a

g

fd


1. The resulting substring abcdefg abfgdcecovers all non-edges incident on c.


e

b

c

a

g

fd


1. The resulting substring abcdefg abfgdcecovers all non-edges incident on c.

2. For this graph it would suffice to repeat this for f: abfgcde abcdefgplus one round for d: dabcdfg

3. Final string:


e

b

c

a

g

fd

abcdefg abfgdce abfgcde abcdefg dabcdfg

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.

1

2

3

4

is permutationally representable (13243142)

Lemma (Kitaev and Seif). A graph is permutationally representable iff it is transitively orientable, i.e. if it is a comparability graph.

Shortcut – a type of digraph

• Acyclic, non-transitive• Contains directed cycle

a, b, c, d, except last edge is reversed

• Non-transitive Not representable


d

b

c

a Missing!

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


Sketch of our algorithm

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


b c

d

a




b c

d

a

b c

d

a



• 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


b c

d

a

b c

d

a

a b c a d c b dExample:

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

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


Corollary: 3-colorable graphs

• 3-colorable graphs are representable

• Red->Green->Blue orientation is shortcut-free!Sergey Kitaev 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.

Construction of non-representable graphs


1. Take a graph that is not a comparability graph (C5 is the smallest example);

2. Add a vertex adjacent to every node of the graph;3. Add other vertices and edges incident to them (optional).

W5 – the smallest non-representable graph

All odd wheels W2t+1 for t ≥ 2are non-representable graphs.

Small non-representable graphs


Relationships of graph classes


Representable

Circle 2-repres.

3-colorable Comparability

Bipartite2-inductive

Partial 2-trees

Outerplanar

2-outerplanar

3-trees

Trees

Chordal

2-trees

Perfect

Planar

4-colorable & K4-free

Split


A property of representable graphs

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

Natural question: Is the converse statement true?

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

Main means of showing non-representability

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


co-T2 T2



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.


One more result

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

Coctail party graph:

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


• 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?


Open problems


Resolved question

Is the Petersen’s graph representable?


Resolved question

Is the Petersen’s graph representable?

It is 3-representable:

1

2

34

9 8

7

6

5 10

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


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?


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?


Thank you for your attention!

THE END

Graphs represented by words

Documents

Transcript of Graphs represented by words