1 / 33

Representable graphs

Representable graphs. Sergey Kitaev Reykjavík University. This is a joint work with. Artem Pyatkin. Sobolev Institute of Mathematics. Application of combinatorics on words to algebra. A semigroup is a set S of elements a , b , c , ... in which an

corby
Download Presentation

Representable graphs

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

  2. 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). Yes, it does! Does S have an infinite number of elements? Thue (1906) Morse (1938) Arshon (1937) Representable Graphs

  3. The Perkins semigroup A monoid is a semigroup Swith an identity element 1, satisfying 1●a=a●1=a for all a in S. 1 The six-element monoid B2, 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 ( ) ( ) ( ) ( ) ( ) ( ) 0= 1= a= a’= aa’= a’a= The Perkins semigroup has played a central role in semigroup theory, particularly as a source of examples and counterexamples. Representable Graphs

  4. 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=w1w2...wkthen the evaluation ofw undere is e(w)=e(w1)e(w2)...e(wk). 1 If w=x2x1x2 and the evaluation e: Var(w)={x1,x2} → B2 is given by e(x1)=a’ and e(x2)=a, we have e(w)=aa’a=a. 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. Representable Graphs

  5. The word problem for a semigroup For example, a semigroup S is commutative iff x1x2 ≈S x2x1. 1 Perkins proved that there exists no finite set of identities of B2 from which allB2-identities can be derived. 1 The word problem for a semigroupS: 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. Representable Graphs

  6. Alternation word digraphs U→ V is an arc in the graph if U and V alternate in the word starting with an element from U 134 234 123 124 12 23 14 13 x1x2x3x1x4 34 24 Alt(x1x2x3x1x4) 1 3 2 4 the level of interest Representable Graphs

  7. 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 Representable Graphs

  8. 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 Representable Graphs

  9. 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) Viffx 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. Representable Graphs

  10. 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 Representable Graphs

  11. 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 Representable Graphs

  12. Properties of representable graphs y x x G G is representable If is representable, then ...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. Representable Graphs

  13. 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=w1xiw2xi+1w3 represents G and xi and xi+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 w2 exactly once. Representable Graphs

  14. 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.) Representable Graphs

  15. 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 ABiff they alternate in BA. (xyxy...xy and yxyx...yx are the only possible outcomes.) Let G1 and G2 be k-representable. Then H1 and H2 are also k-representable (see the picture below). H2 H1 G1 G2 G1 G2 x y x=y Representable Graphs

  16. Properties of representable graphs Constructions for the case k=3: H1 w1=A1xA2xA3x represents G1 G1 G2 x y w2=yB1yB2yB3represents G2 w3=A1xA2yxB1A3yxB2yB3represents H1 H2 w4=A1zA2B1zA3B2zB3represents H2 G1 G2 x=y=z Representable Graphs

  17. 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 at least one of its possible orientations is a comporability graph of a poset. In particular, all bipartite graphs are permutationally representable. Representable Graphs

  18. 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 P1P2...Pk permut. represents H then P1xP2x...Pkx represents G. If A1xA2x...AkxAk+1 represents G then each Ai must be a permutation since x is adjacent to each vertex. Now, the word (A1\A0)A0A1...AkAk+1(Ak\Ak+1) permutationally represents H. The lemmas give us a method to construct non-representable graphs. Representable Graphs

  19. 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. Representable Graphs

  20. Small non-representable graphs Representable Graphs

  21. 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? Representable Graphs

  22. 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. Representable Graphs

  23. 2-representable graphs The graph below is representable but not 2-representable. 3 4 2 1 5 6 7 8 Home assignment: Prove it! Representable Graphs

  24. 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 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. y 2-representable and thus 3-representable Representable Graphs

  25. 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. y 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 v u q The same story with adding v, and t ... x z t Ups, we have got a non-representable graph! Representable Graphs

  26. 3-representable graphs An example of applying the construction in the lemma. 3 4 2 1 5 6 7 8 A 2-uniform word representing the cycle (134265): 314324625615 Make it 3-uniform: 314265314324625615 Apply the construction in the lemma: 378174265314387284625615 Representable Graphs

  27. 3-representable graphs graph G graph G1 is a subdivision of G (replacing edges by simple paths) G is a minor of G1 and G2 graph G2 is the 3-subdivision of G Representable Graphs

  28. 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 x1, x2, ..., xk. Then x1x2...xkxkxk-1...x1x1x2...xk 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. Representable Graphs

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

  30. 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? Representable Graphs

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

  32. 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? Representable Graphs

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

More Related