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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  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 G + H = G H G & H = G H Graphs represented by words

  10. 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

  11. 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

  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 Graphs 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 FA B E uv  E  u v E A F Graphs represented by words

  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 FA B E E A F Graphs represented by words

  15. 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

  16. 2-representable graphs A B C D E F G H C D H G F A B E Circle graphs Graphs represented by words

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

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

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

  20. 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

  21. 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

  22. Representing comparability graphs • The resulting substringabcdefg abfgdcecoversall non-edges incident on c. a b c d f e g Graphs represented by words

  23. 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

  24. 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

  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 a Missing! b c d Graphs represented by words

  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 Graphs represented by words

  27. 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

  28. 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

  29. 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

  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 Graphs represented by words

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

  32. 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

  33. 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

  34. Small non-representable graphs Graphs represented by words

  35. 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

  36. 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

  37. A non-representable graph whose induced neighborhood graphs are all comparability co-T2 T2 Graphs represented by words

  38. 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

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

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

  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 Open! Graphs represented by words

  42. 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

  43. Resolved question Is the Petersen’s graph representable? Yes! Graphs represented by words

  44. 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

  45. 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

  46. 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

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