1 / 47

Graphs represented by words

Graphs represented by words. Joint work with. Magnus M. Halldorsson. Sergey Kitaev Reykjavik University. Reykjavik University. Artem Pyatkin. Sobolev Institute of Mathematics. Basic definitions. A finite word over { x , y } is alternating if it does not contain xx and yy.

fergus
Download Presentation

Graphs represented by words

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related