Bi-Point-Determining Graphs

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A bijection for counting bi-point-determining graphs using the combinatorial theory of species.

23 pages, Combinatorics Seminar, Brandeis University, 2007.

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Bi-Point-Determining Graphs

  1. 1. Counting Bi-Point-Determining Graphs A Bijection Ji Li Combinatorics Seminar Department of Mathematics Brandeis University April 20th, 2007
  2. 2. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsOutline1 Point-Determining Graphs2 Bi-Point-Determining Graphs3 A Bijection for Bi-Point-Determining Graphs4 Generating FunctionsJ. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 2 / 23
  3. 3. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsNeighborhood of a VertexDefinitionIn a graph G , the neighborhood of a vertex v is the set of vertices adjacent to v ,the augmented neighborhood of a vertex is the union of the vertex itself and itsneighborhood.Example v w1 w2 w3 w4In the above figure, the neighborhood of vertex v is the set {w1 , w2 , w3 , w4 }, whilethe augmented neighborhood of v is the set {v , w1 , w2 , w3 , w4 }.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 3 / 23
  4. 4. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsPoint-Determining Graphs andCo-Point-Determining GraphsDefinition • A graph is called point-determining if no two vertices of this graph have the same neighborhoods. • A graph is called co-point-determining if its complement is point-determining. • Equivalently, a graph is co-point-determining if no two vertices of this graph have the same augmented neighborhoods.ExampleThe graph on the left is co-point-determining, and the graph on the right ispoint-determining. These two graphs are complements of each other.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 4 / 23
  5. 5. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsA TransformationA transformation from a graph G to a point-determining graph P. 3 3 9 2 9 2 1 5 1 5 8 6 8 6 4 7 4 7The above figure illustrates the transformation from a graph G with vertex set [11]to a point-determining graph P with vertex set {{1, 9, 3}, {8}, {4, 7}, {6}, {2, 5}}.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 5 / 23
  6. 6. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsAnother TransformationA transformation from a graph G to a co-point-determining graph Q. 3 3 9 2 9 2 1 5 1 5 8 6 8 6 4 7 4 7 Here is another similar transformation from a graph G with vertex set [11] to a co-point-determining graph Q with vertex set {{1, 9, 3}, {8}, {4, 7}, {6}, {2, 5}}.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 6 / 23
  7. 7. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsOutline1 Point-Determining Graphs2 Bi-Point-Determining Graphs3 A Bijection for Bi-Point-Determining Graphs4 Generating FunctionsJ. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 7 / 23
  8. 8. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsBi-Point-Determining GraphsDefinitionA bi-point-determining graph is a graph that is both point-determining andco-point-determining.ExampleListed in above are all unlabeled bi-point-determining graphs with no more than 5vertices.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 8 / 23
  9. 9. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsTransform a Graph into a Bi-Point-Determining Graph?Yes. 6 3 6 3 8 8 5 5 1 1 2 2 4 7 4 7 On each step, we group vertices with the same neighborhoods or the same augmented neighborhoods, and get a new graph whose vertices are sets 6 3 of vertices of the previous graph. 8 In this way, we end up with a graph in which no two vertices have the same neighborhood 5 or the same augmented neighborhood, i.e., 1 a bi−point−determining graph. 2 4 7 But, we would like to keep track of the procedure so that the original graph can be reconstructed....J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 9 / 23
  10. 10. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsAlternating Phylogenetic TreesDefinition • A phylogenetic tree is a rooted tree with labeled leaves and unlabeled internal vertices in which no vertex has exactly one child. • An alternating phylogenetic tree is either a single vertex, or a phylogenetic tree with more than one labeled vertex whose internal vertices are colored black or white, where no two adjacent vertices are colored the same way.Example 5 4 8 An alternating 6 9 phylogenetic tree on 9 vertices, where the root 3 1 is colored black. 7 2J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 10 / 23
  11. 11. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsOutline1 Point-Determining Graphs2 Bi-Point-Determining Graphs3 A Bijection for Bi-Point-Determining Graphs4 Generating FunctionsJ. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 11 / 23
  12. 12. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsAn Informal DescriptionThe claim is —The structure of alternating phylogenetic trees can be used to keep track of thetransformation of an arbitrary graph into a bi-point-determining graph. We let An illustration • G denote arbitrary graphs S • R denote arbitrary S G bi-point-determining R S graphs • S denote alternating phylogenetic trees.The above figure means..the structure of graphs is the structure of bi-point-determining graphssuperimposed with the structures of alternating phylogenetic trees.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 12 / 23
  13. 13. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsAn Illustration of the BijectionExample V1 V2 6 3 V3 V4 8 5 3 1 6 8 V1 V2 2 4 7 7 2 5 1 4 V3 V4J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 13 / 23
  14. 14. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsThe Formal Description of the BijectionFor any finite set U, we construct a bijection between the set of triples of the form (π, R, γ) such that • π is a partition of the set U, i.e., the set of and π = {V1 , V2 , . . . , Vk } graphs with • R is a bi-point-determining graph with vertex set π vertex set U • γ is a set of alternating phylogenetic trees {S1 , S2 , . . . , Sk }, where each Si is an alternating phylogenetic tree with vertex set Vi .Next, we will see.. • how to get a triple from an arbitrary graph • how to construct a graph from a given tripleJ. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 14 / 23
  15. 15. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsFrom a Graph to a TripleExample 6 3 6 3 8 8 5 5 1 2 4 7 1 2 4 7 6 8 2 5 1 4 On each step, we group vertices with the same neighborhoods or vertices with the same augmented neighborhods. 6 3 Whenever vertices with the same neighborhods are grouped, 8 we connect the corresponding vertices/alternating phylogenetic 5 trees with a black node. 1 2 4 7 Whenever vertices with the same augmented neighborhoods are grouped, we connected the corresponding vertices/ 7 alternating phylogenetic trees with a white node. 6 8 2 5 1 4J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 15 / 23
  16. 16. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsFrom a Triple to a Graph Example 3 V1 V2 V1 6 8 V2 Given a triple (π, R, γ), where V3 V4 7 • π = {V1 , V2 , . . . } is a 2 5 V3 1 4 V4 partition of U • R is a bi-point-determining graph on the blocks of π 6 3 • γ is a set {S1 , S2 , . . . } in 8 which each Si is an 5 alternating phylogenetic 1 2 tree labeled on the set Vi . 4 7J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 16 / 23
  17. 17. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsFrom a Triple to a Graph: Continue Then there is a unique graph G Example with vertex set U such that... 3 Vertices v1 and v2 of G are V1 V2 6 8 V1 V2 adjacent if and only if exactly V3 V4 7 one of the following two V3 1 4 V4 2 5 conditions is satisfied: a) v1 and v2 are labels of vertices of Si for some i, 6 3 and the common ancestor 8 of v1 and v2 in Si is colored 5 white. 1 2 b) v1 ∈ Vi , v2 ∈ Vj , and Vi and Vj are adjacent vertices 4 7 in the bi-point-determining graph R.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 17 / 23
  18. 18. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsThe Common AncestorDefinitionThe common ancestor of two vertices a and b in a phylogenetic tree is defined tobe such that if we take the unique shortest path from a to b, say, w0 w1 · · · wl ,with w0 = a and wl = b, then the common ancestor of a and b is the unique wifor which both wi −1 and wi +1 are children of wi .Example 5 4 8 6 9 3 1 7 2The common ancestor of vertices 5 and 4 is colored black, while the commonancestor of vertices 5 and 3 is colored white.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 18 / 23
  19. 19. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsOutline1 Point-Determining Graphs2 Bi-Point-Determining Graphs3 A Bijection for Bi-Point-Determining Graphs4 Generating FunctionsJ. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 19 / 23
  20. 20. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsThis Bijection Gives Rise to Functional EquationsAs illustrated.. S S G We get R S G = R ◦ S.Which reads..The structure of graphs is the structure of bi-point-determining graphs composedwith the structure of alternating phylogenetic trees.J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 20 / 23
  21. 21. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsThe Exponential Generating Functionfor Labeled Bi-Point-Determining GraphsWe write • R(x) to be the exponential generating function of labeled bi-point-determining graphs; • G (x) to be the exponential generating function of labeled graphs, which is given by n xn G (x) = 2 2 . n! n≥0Then we get R(x) = G (2 log(1 + x) − x). 4 x x x5 x6 x7 x8 R(x) = + 12 + 312 + 13824 + 1147488 + 178672128 + ··· . 1! 4! 5! 6! 7! 8!J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 21 / 23
  22. 22. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsThe Ordinary Generating Functionfor Unlabeled Bi-Point-Determining GraphsWe writeR(x) to be the (ordinary) generating function of unlabeled bi-point-determininggraphs. Then R(x) = ZG (x − 2x 2 , x 2 − 2x 4 , . . . ).Here ZG is the so-called cycle index of the structures of graphs G , and theformula for ZG is known. We write down the beginning Compare with.. terms of R(x) R(x) = x + x 4 + 6x 5 + 36x 6 + 324x 7 + 5280x 8 + ··· .J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 22 / 23
  23. 23. Point-Determining Graphs Bi-Point-Determining Graphs A Bijection for Bi-Point-Determining Graphs Generating FunctionsThe EndThank you for your patience!J. L. (Brandeis Combinatorics Seminar) Counting Bi-Point-Determining Graphs 04. 20. 2007 23 / 23

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