representable graphs sergey kitaev reykjavík university sobolev institute of mathematics this is a...

Representable graphs

Sergey KitaevReykjavík University

Sobolev Institute of Mathematics

This is a joint work with

Artem Pyatkin

Sergey Kitaev Representable Graphs

Application of combinatorics on words to algebra

A semigroup is a set S of elements a, b, c, ... in which anassociative operation ● is defined.

The element z is a zero element if z●a=a●z=z for all a in S.

Let S be a semigroup generated by three elements, such that the square of every element in S is zero (thus a●a=z for all a in S).

Does S have an infinite number of elements?

Thue (1906)

Arshon (1937)

Morse (1938)

Yes, it does!


The Perkins semigroup

A monoid is a semigroup S with an identity element 1, satisfying1●a=a●1=a for all a in S.

The six-element monoid B2, the Perkins semigroup, consists of the following six two-by-two matrices:

0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0

1

0= 1= a’=a= a’a=aa’=( () )))) )((((

The Perkins semigroup has played a central role in semigroup theory,particularly as a source of examples and counterexamples.


The word problem for a semigroup

Var(w) denotes the letters occurring in a word w.

If K contains Var(w) and S is a semigroup, then an evaluation is afunction e: K → S. If w=w1w2...wk then the evaluation of w under e is e(w)=e(w1)e(w2)...e(wk).

If w=x2x1x2 and the evaluation e: Var(w)={x1,x2} → B2 is given bye(x1)=a’ and e(x2)=a, we have e(w)=aa’a=a.

1

If for all evaluations e: Var(u) U Var(v) → S we have e(u)=e(v), thenthe words u and v are said to be S-equivalent (denoted u ≈S v) and u ≈S v is said to be an identity of S.


The word problem for a semigroup

For example, a semigroup S is commutative iff x1x2 ≈S x2x1.

Perkins proved that there exists no finite set of identities of B2

from which all B2-identities can be derived.

1

1

The word problem for a semigroup S: Given two words u, v, is u ≈S v?

For a finite semigroup, the word problem is decidable, but the computational complexity of the word problem (the term-equivalence problem) is generally difficult to classify.


Alternation word digraphs

x1x2x3x1x4

Alt(x1x2x3x1x4)1 3

2 4

3424

1423

1312

123

134 234

124

U → V is an arc in the graph if U and V alternatein the word starting withan element from U

the level of interest


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


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) V 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


What is coming next ...

Some properties of the representable graphs Examples of non-representable graphs Some classes of 2- and 3-representable

graphs Open problems


Properties of representable graphs

G

x

IfG

x

is representable, then

y

is representable

...x...x...x...x...x... ...yxy...x...yxy...x...yxy...

Corollary. All trees are (2-)representable.

More generally, all graphs having at most 3 cycles are representable.



If G is (k-)representable and G’ is an induced subgraph of G then G’ is also (k-)representable. (The class of (k-)representable graphs is hereditary.)

If w represents G=(V,E) and XV, then w\X represents G’ on V\X.

If w=w1xiw2x

i+1w3 represents G and xi and xi+1 are two consecutiveoccurrences of a letter x, then all possible candidates for the vertex x to be adjacent to in G are among the letters appearing in w2 exactly once.



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

Let w be a k-uniform word representing G.P(w) is the permutation obtained by removing all but the first(leftmost) occurrences of the letters of w (the initial permutation).Then P(w)w is a (k+1)-uniform word representing G.

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 w=AB is a k-uniform word representing G then w’=BA k-represents G.

x and y alternate in AB iff they alternate in BA. (xyxy...xy and yxyx...yxare the only possible outcomes.)

Let G1 and G2 be k-representable. Then H1 and H2 are also k-representable (see the picture below).

x y

H1H2

x=y

G1 G2G2 G1



Constructions for the case k=3:

x y

H1

G1 G2

H2

x=y=z

G2G1

w1=A1xA2xA3x represents G1

w2=yB1yB2yB3 represents G2

w3=A1xA2yxB1A3yxB2yB3 represents H1

w4=A1zA2B1zA3B2zB3 represents H2



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.

Lemma (Kitaev and Seif). A graph is permutationally representable iff atleast one of its possible orientations is a comporability graph of a poset.In particular, all bipartite graphs are permutationally representable.

1

2

3

4

is permutationally representable (13243142)


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.

Proof. If P1P2...Pk permut. represents H then P1xP2x...Pkx represents G.

If A1xA2x...AkxAk+1 represents G then each Ai must be a permutation since x is adjacent to each vertex. Now, the word (A1\A0)A0A1...AkAk+1(Ak\Ak+1) permutationally represents H.

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


A property of representable graphs

For a vertex x, N(x) denotes the set of all the neighbors of x in a graph.

Theorem. If G=(V,E) is representable then for every x V the graph induced by N(x) is permutationally representable.

Open problem: Is the opposite statement true?


2-representable graphs

If w=AxBxC is a 2-uniform word representing a graph G then x is adjacent to those and only those vertices in G that occurs exactly once in B.

A graph is outerplanar if it can be drawn in the plane in such a way that no two edges meet in a point other than a common vertex and all the vertices lie in the outer face.

Odd wheels on at least 6 nodes, being planar, are not representable.

Theorem. If a graph is outerplanar then it is 2-representable.



The graph below is representable but not 2-representable.

1

3 4

25 6

7 8

Home assignment: Prove it!



Lemma. Let G be a 3-representable graph and x and y are verticesof it. Denote by H the graph obtained from G by adding to it a path of length at least 3 connecting x and y. Then H is 3-representable.

x

y

2-representable and thus 3-representable

also 3-representable

Idea of the proof: Reduce to the case ofadding just two nodes u and v, and substitute certain x in a word-representant of G by uxvu and certain y by vuyv.



Lemma. Let G be a 3-representable graph and x and y are verticesof it. Denote by H the graph obtained from G by adding to it a path of length at least 3 connecting x and y. Then H is 3-representable.

x

y

zq

u v

t

3 is essential here

the complete graph is 3-represented by xyzqxyzqxyzq

If 3 could be changed by 2 in the lemma thenadding u would montain 3-representability

The same story with adding v, and t ...

Ups, we have got a non-representable graph!



1

3 4

25 6

7 8

An example of applying the construction in the lemma.

A 2-uniform word representing the cycle (134265): 314324625615

Make it 3-uniform: 314265314324625615

Apply the construction in the lemma: 378174265314387284625615



graph G

graph G1 is a subdivision of G (replacing edges by simple paths)

graph G2 is the 3-subdivision of G

G is a minor of G1 and G2



Theorem. For every graph G there exists a 3-representablegraph H that contains G as a minor. In particular, a 3-subdivisionof every graph G is 3-representable.

Proof. Suppose the nodes of G are x1, x2, ..., xk. Then x1x2...xkxkxk-1...x1x1x2...xk 3-represents the graph withno edges on the nodes. Now, for each pair of nodes x and y, add a simple path of length 3 between x and y If there is an edge between x and y in G; otherwise don’t do anything. We are done by the lemma.



examples of prisms

Theorem. Every prism is 3-representable.


Open problems

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

graphs? Are there non-representable graphs of maximum

degree 3? Are there 3-chromatic non-representable graphs?


Open problems

Is the Petersen’s graph representable?


Open 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

representable graphs sergey kitaev reykjavík university sobolev institute of mathematics this is a...

Documents

word w

letters x

xyzazxayxzyax x

letter x

vertex x

indicated x

representable iff g

y v iff x