1. Representable graphs Sergey Kitaev Reykjavík University Sobolev Institute of Mathematics This is a joint work with Artem Pyatkin

2. Sergey Kitaev Representable Graphs Application of combinatorics on words to algebra A semigroup is a set S of elements a, b, c,... in which an associative 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!

3. Sergey Kitaev Representable Graphs The Perkins semigroup A monoid is a semigroup S with an identity element 1, satisfying 1●a=a●1=a for all a in S. The six-element monoid B 2, 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=0=1=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.

4. Sergey Kitaev Representable Graphs 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 a function e: K → S. If w=w 1 w 2...w k then the evaluation of w under e is e(w)=e(w 1 )e(w 2 )...e(w k ). If w=x 2 x 1 x 2 and the evaluation e: Var(w)={x 1,x 2 } → B 2 is given by e(x 1 )=a’ and e(x 2 )=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), then the 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.

5. Sergey Kitaev Representable Graphs The word problem for a semigroup For example, a semigroup S is commutative iff x 1 x 2 ≈ S x 2 x 1. Perkins proved that there exists no finite set of identities of B 2 from which all B 2 -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.

6. Sergey Kitaev Representable Graphs Alternation word digraphs x1x2x3x1x4x1x2x3x1x4 Alt(x 1 x 2 x 3 x 1 x 4 ) 1 3 24 34 24 14 23 13 12 123 134234 124 U → V is an arc in the graph if U and V alternate in the word starting with an element from U the level of interest

7. Sergey Kitaev Representable Graphs 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

8. Sergey Kitaev Representable Graphs 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

9. Sergey Kitaev Representable Graphs 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, and 2.(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.

10. Sergey Kitaev Representable Graphs Example of a representable graph cycle graph x y v za xyzxazvay represents the graph xyzxazvayv 2-represents the graph Switching the indicated x and a would create an extra edge

11. Sergey Kitaev Representable Graphs 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

12. Sergey Kitaev Representable Graphs Properties of representable graphs G x If G 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.

13. Sergey Kitaev Representable Graphs Properties of representable graphs 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 X  V, then w\X represents G’ on V\X. If w=w 1 x i w 2 x i+1 w 3 represents G and x i and x i+1 are two consecutive occurrences of a letter x, then all possible candidates for the vertex x to be adjacent to in G are among the letters appearing in w 2 exactly once.

14. Sergey Kitaev Representable Graphs Properties of representable graphs 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 iff G and H are representable. (Take concatenation of the corresponding words representants having at least two copies of each letter.)

15. Sergey Kitaev Representable Graphs Properties of representable graphs 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...yx are the only possible outcomes.) Let G 1 and G 2 be k-representable. Then H 1 and H 2 are also k-representable (see the picture below). xy H1H1 H2H2 x=y G1G1 G2G2 G2G2 G1G1

16. Sergey Kitaev Representable Graphs Properties of representable graphs Constructions for the case k=3: xy H1H1 G1G1 G2G2 H2H2 x=y=z G2G2 G1G1 w 1 =A 1 xA 2 xA 3 x represents G 1 w 2 =yB 1 yB 2 yB 3 represents G 2 w 3 =A 1 xA 2 yxB 1 A 3 yxB 2 yB 3 represents H 1 w 4 =A 1 zA 2 B 1 zA 3 B 2 zB 3 represents H 2

17. Sergey Kitaev Representable Graphs Properties of representable graphs A graph is permutationally representable if it can be represented by a word of the form P 1 P 2...P k where P i s are permutations of the same set. Lemma (Kitaev and Seif). A graph is permutationally representable iff at least 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)

18. Sergey Kitaev Representable Graphs 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 P 1 P 2...P k permut. represents H then P 1 xP 2 x...P k x represents G. If A 1 xA 2 x...A k xA k+1 represents G then each A i must be a permutation since x is adjacent to each vertex. Now, the word (A 1 \A 0 )A 0 A 1...A k A k+1 (A k \A k+1 ) permutationally represents H. The lemmas give us a method to construct non-representable graphs.

19. Sergey Kitaev Representable Graphs Construction of non-representable graphs 1.Take a graph that is not a comparability graph (C 5 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). W 5 – the smallest non-representable graph All odd wheels W 2t+1 for t ≥ 2 are non-representable graphs.

20. Sergey Kitaev Representable Graphs Small non-representable graphs

21. Sergey Kitaev 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?

22. Sergey Kitaev Representable Graphs 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.

23. Sergey Kitaev Representable Graphs 2-representable graphs The graph below is representable but not 2-representable. 1 34 2 56 78 Home assignment: Prove it!

24. Sergey Kitaev Representable Graphs 3-representable graphs Lemma. Let G be a 3-representable graph and x and y are vertices of 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 of adding just two nodes u and v, and substitute certain x in a word-representant of G by uxvu and certain y by vuyv.

25. Sergey Kitaev Representable Graphs 3-representable graphs Lemma. Let G be a 3-representable graph and x and y are vertices of 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 z q 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 then adding u would montain 3-representability The same story with adding v, and t... Ups, we have got a non-representable graph!

26. Sergey Kitaev Representable Graphs 3-representable graphs 1 34 2 56 78 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

27. Sergey Kitaev Representable Graphs 3-representable graphs graph G graph G 1 is a subdivision of G (replacing edges by simple paths) graph G 2 is the 3-subdivision of G G is a minor of G 1 and G 2

28. Sergey Kitaev Representable Graphs 3-representable graphs Theorem. 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. Proof. Suppose the nodes of G are x 1, x 2,..., x k. Then x 1 x 2...x k x k x k-1...x 1 x 1 x 2...x k 3-represents the graph with no 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.

29. Sergey Kitaev Representable Graphs 3-representable graphs examples of prisms Theorem. Every prism is 3-representable.

30. Sergey Kitaev Representable Graphs 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?

31. Sergey Kitaev Representable Graphs Open problems Is the Petersen’s graph representable?

32. Sergey Kitaev Representable Graphs 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?

33. Sergey Kitaev Representable Graphs Thank you for your attention! THE END