1. Graphs represented by words Sergey Kitaev Reykjavik University Sobolev Institute of Mathematics Joint work with Artem Pyatkin Magnus M. Halldorsson Reykjavik University

2. Basic definitions Sergey KitaevGraphs 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

3. Basic definitions Sergey KitaevGraphs 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 x and y do not alternate in w = xyzazyaxyxzyax

4. Basic definitions Sergey KitaevGraphs represented by words 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)  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.

5. Example of a representable graph Sergey KitaevGraphs represented by words 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

6. Sergey KitaevGraphs represented by words Cliques and Independent Sets KnKn Clique KnKn Independent set W=ABC...Z ABC...Z W=ABC...YZ ZY...CBA V={A,B,C,...Z}

7. Sergey KitaevGraphs 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.

8. Sergey KitaevGraphs 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.

9. 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 Sergey KitaevGraphs represented by words G H += H G G H &= G H

10. Operations Not Preserving Representability Taking the line graph Taking the complement Attaching two graphs at more than 1 node Sergey KitaevGraphs represented by words 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.

11. Properties of representable graphs Sergey KitaevGraphs represented by words 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 iff G 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.

12. 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 Sergey KitaevGraphs represented by words

13. 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 F A B E Sergey KitaevGraphs represented by words E A F u v uv  E 

14. 2-representable graphs 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 Sergey KitaevGraphs represented by words E A F

15. 2-representable graphs Sergey KitaevGraphs represented by words

16. 2-representable graphs Sergey KitaevGraphs represented by words  Circle graphs

17. Comparability graphs We can orient the edges to form a transitive digraph They correspond to partial orders. Sergey KitaevGraphs represented by words

18. Comparability graphs We can orient the edges to form a transitive digraph They correspond to partial orders. Sergey KitaevGraphs represented by words

19. Comparability graphs We can orient the edges to form a transitive digraph They correspond to partial orders. Sergey KitaevGraphs represented by words

20. Representing comparability graphs 1.Form a topological ordering, where a given letter, say c, is as early as possible: abcdefg Sergey KitaevGraphs represented by words e b c a g f d

21. Representing comparability graphs 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 Sergey KitaevGraphs represented by words e b c a g f d

22. Representing comparability graphs 1.The resulting substring abcdefg abfgdce covers all non-edges incident on c. Sergey KitaevGraphs represented by words e b c a g f d

23. Representing comparability graphs 1.The resulting substring abcdefg abfgdce covers all non-edges incident on c. 2.For this graph it would suffice to repeat this for f: abfgcde abcdefg plus one round for d: dabcdfg 3.Final string: Sergey KitaevGraphs represented by words e b c a g f d abcdefg abfgdce abfgcde abcdefg dabcdfg

24. Properties of representable graphs Sergey KitaevGraphs represented by words 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. 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.

25. Shortcut – a type of digraph Acyclic, non-transitive Contains directed cycle a, b, c, d, except last edge is reversed Non-transitive  Not representable Sergey KitaevGraphs represented by words d b c a Missing!

26. 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 Sergey KitaevGraphs represented by words

27. Sketch of our algorithm Chain together copies of the digraph (= D’) – If ab  D, then b i a i+1  D’ Sergey KitaevGraphs represented by words bc d a

28. Sketch of our algorithm Chain together copies of the digraph (= D’) – If ab  D, then b i a i+1  D’ Sergey KitaevGraphs represented by words bc d a bc d a

29. Sketch of our algorithm Chain together copies of the digraph (= D’) – If ab  D, then b i a i+1  D’ Form a topsort of D’ of pairs of copies. – In 1 st copy, some letter d occurs as late as possible – In 2 nd copy d occurs as early as possible Sergey KitaevGraphs represented by words bc d a bc d a a b cadc b dExample: We allow the topsort to traverse the 2 nd copy before finishing the 1 st. The added edges ensure that adjacent nodes still alternate.

30. 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 Sergey KitaevGraphs represented by words

31. Corollary: 3-colorable graphs 3-colorable graphs are representable Red->Green->Blue orientation is shortcut-free! Sergey KitaevGraphs represented by words

32. Non-representable graphs Sergey KitaevGraphs represented by words 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.

33. Construction of non-representable graphs Sergey KitaevGraphs represented by words 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.

34. Small non-representable graphs Sergey KitaevGraphs represented by words

35. Relationships of graph classes Sergey KitaevGraphs represented by words Representable Circle  2-repres. 3-colorableComparability Bipartite 2-inductive Partial 2-trees Outerplanar 2-outerplanar 3-trees Trees Chordal 2-trees Perfect Planar 4-colorable & K 4 -free Split

36. Sergey KitaevGraphs represented by words 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

37. A non-representable graph whose induced neighborhood graphs are all comparability Sergey KitaevGraphs represented by words co-T 2 T2T2

38. 3-representable graphs Sergey KitaevGraphs represented by words 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.

39. Sergey KitaevGraphs represented by words One more result We can construct graphs with represntation number k=[n/2] Coctail party graph:

40. Sergey KitaevGraphs represented by words One more result We can construct graphs with represntation number k=[n/2] Coctail party graph:

41. 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 Sergey KitaevGraphs represented by words

42. 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? Sergey KitaevGraphs represented by words Open problems

43. Sergey KitaevGraphs represented by words Resolved question Is the Petersen’s graph representable?

44. Sergey KitaevGraphs represented by words Resolved question Is the Petersen’s graph representable? It is 3-representable: 1 2 3 4 98 7 6 510 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

45. Sergey KitaevGraphs 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?

46. Sergey KitaevGraphs 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?

47. Sergey KitaevGraphs represented by words Thank you for your attention! THE END