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## Graphs represented by words

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**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. 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**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**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**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**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**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**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**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**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**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**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**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**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**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**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**Comparability graphs**• We can orient the edges to form a transitive digraph • They correspond to partial orders. Graphs represented by words**Comparability graphs**• We can orient the edges to form a transitive digraph • They correspond to partial orders. Graphs represented by words**Comparability graphs**• We can orient the edges to form a transitive digraph • They correspond to partial orders. Graphs represented by words**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**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**Representing comparability graphs**• The resulting substringabcdefg abfgdcecoversall non-edges incident on c. a b c d f e g Graphs represented by words**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**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**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**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**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**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**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**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**Corollary: 3-colorable graphs**• 3-colorable graphs are representable • Red->Green->Blue orientation is shortcut-free! Graphs represented by words**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**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**Small non-representable graphs**Graphs represented by words**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**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**A non-representable graph whose induced neighborhood graphs**are all comparability co-T2 T2 Graphs represented by words**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**One more result**We can construct graphs with represntation number k=[n/2] Coctail party graph: Graphs represented by words**One more result**We can construct graphs with represntation number k=[n/2] Coctail party graph: Graphs represented by words**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**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**Resolved question**Is the Petersen’s graph representable? Yes! Graphs represented by words**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**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**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**Thank you for your attention!**THE END Graphs represented by words