Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs. Maryam Bahrani Jérémie Lumbroso

Transcription

1 Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs Maryam Bahrani Jérémie Lumbroso

2 1. Preliminaries 2. Results Outline Symbolic Method on Trees Graph Decomposition Cactus Graphs Previous work on Enumerating of Cacti 3. Methodoloy Grammar Templates Random Generation Enumeration The Split Decomposition Characterization and Grammar Unrooting Random Sampling

3 Symbolic Method on Trees T T A binary tree is either a leaf or an internal node, and a left subtree, and a right subtree. T = [ ( T T ) symbolic specification T (z) =1+T (z) z T (z) generating function equation Number of binary trees with 4 internal nodes exact enumeration

4 Decomposing General Graphs Goal Apply the symbolic method to general graphs Approach Tree Decomposition: Converts graphs to trees modular decomposition split decomposition

5 Decomposing General Graphs Goal Apply the symbolic method to general graphs Approach Tree Decomposition: Converts graphs to trees modular decomposition split decomposition

6 Non-Decomposable Graphs Previous Work Studied several subsets of fully decomposable graphs (with respect to the split decomposition) relied on full decomposability to prove results Decomposition base cases: cliques stars primes subsets of Motivating Question Can we study graphs that are not fully decomposable? Must handle prime nodes Start with manageable prime nodes e.g. cycles cactus graphs!

7 Cactus Graphs A graph is a cactus iff every edge is part of at most one cycle. not cactus cactus

8 Cactus Graphs A graph is a cactus iff every edge is part of at most one cycle. not cactus pure 3-cactus mixed cactus

9 Cactus Graphs A graph is a cactus iff every edge is part of at most one cycle. not cactus pure 3-cactus unlabeled cactus labeled cactus mixed cactus

10 Cactus Graphs A graph is a cactus iff every edge is part of at most one cycle. not cactus pure 3-cactus unlabeled cactus labeled cactus mixed cactus from Enumeration of m-ary Cacti (Bóna et al.) plane (vs. free) cactus

11 Cacti in the Wild

12 Prior Enumerative Work On the Number of Husimi Trees Harary and Uhlenbeck (1952): proposed method for enumerating free, unlabeled cacti derived functional equations for 3- and 4-cacti referenced future paper for a more systematic treatment of the general case of pure k-cacti Enumeration of m-ary cacti Miklós Bóna et al. (1999): enumerated pure, plane, unlabeled cacti. only plane cacti complicated methods not easily generalizable hard to extract not symbolic (no random generation) no unified framework for all cacti

13 1. Preliminaries 2. Results Outline Symbolic Method on Trees Graph Decomposition Cactus Graphs Previous work on Enumerating of Cacti 3. Methodoloy Grammar Templates Random Generation Enumeration The Split Decomposition Characterization and Grammar Unrooting Random Sampling

14 Unified Framework Grammar Templates Our Work common rooted Random Generation fast, uniform samplers unrooted A random mixed cactus with 309 vertices and 80 cycles guidelines A random mixed cactus with 933 vertices and 239 cycles

15 1. Preliminaries 2. Results Outline Symbolic Method on Trees Graph Decomposition Cactus Graphs Previous work on Enumerating of Cacti 3. Methodoloy Grammar Templates Random Generation Enumeration The Split Decomposition Characterization and Grammar Unrooting Random Sampling

16 Methodolgy: Overview Boltzmann Sampler split decomposition G = Z (P + S C ) P = Seq =4 (Z + S X ) S X = Z Seq >1 (P) S C = Cyc >2 (P) symbolic specification computer algebra system (CAS) 0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 8, 0, 19, 0, 48, 0, 126, 0, 355, 0, 1037,...

17 Methodology: The Split Decomposition (Gioan and Paul, 2013) Def. A graph-labeled tree is a pair, where is a tree and is a family of graphs, such that Every tree node is labeled with a graph There is exactly one tree-edge for every vertex of

18 Methodology: The Split Decomposition (Gioan and Paul, 2013) Def. A graph-labeled tree is a pair, where is a tree and is a family of graphs, such that Every tree node is labeled with a graph There is exactly one tree-edge for every vertex of Def. A split in a graph is a bipartition of the vertices into two subsets and such that Each side has at least size 2 The edges crossing the bipartition induce a complete bipartite graph.

19 Methodology: The Split Decomposition (Gioan and Paul, 2013) Def. A graph-labeled tree is a pair, where is a tree and is a family of graphs, such that Every tree node is labeled with a graph There is exactly one tree-edge for every vertex of Def. A split in a graph is a bipartition of the vertices into two subsets and such that Each side has at least size 2 The edges crossing the bipartition induce a complete bipartite graph. split join

20 Methodology: The Split Decomposition Decomposition base cases: degenerate nodes: clique K center prime nodes: e.g. cycle P star S extremities

21 Methodology: The Split Decomposition Decomposition base cases: degenerate nodes: clique K center prime nodes: e.g. cycle P star S Theorem (Cunningham 82): The split decomposition tree into prime and degenerate nodes is unique as long as the following conditions are met: extremities Every non-leaf node has degree at least 3 no tree edge links two vertices with clique labels no tree edge links the center of a star to an extremity of a star

22 Methodology: Characterization and Grammar Characterization: Cactus graphs can are in bijection with graph-labeled trees where internal nodes are stars and polygons; no polygons are adjacent; the centers of star nodes are attached to leaves; the extremities of star nodes attached to polygons. This characterization can be captured using a symbolic grammar.

23 Methodology: Characterization and Grammar Grammar (unlabeled free pure k-cacti): k-cactus graph rooted at a vertex distinguished leaf polygon entered from a subtree star entered from an extremity star entered from its center set of n (unordered) elements from A undirected sequence of n elements from A

24 Methodology: Unrooting Subtleties Where do we start decomposing from? unlabeled structures have symmetries different set of symmetries for different starting points ( roots )

25 Methodology: Unrooting Subtleties Where do we start decomposing from? unlabeled structures have symmetries different set of symmetries for different starting points ( roots ) Dissymmetry theorem (Bergeron et al. 98): allows us to correct for symmetries of trees proof by observing that the tree center (midpoint of diameter) is distinguished by definition

26 Methodology: Unrooting Subtleties Where do we start decomposing from? unlabeled structures have symmetries different set of symmetries for different starting points ( roots ) Dissymmetry theorem (Bergeron et al. 98): allows us to correct for symmetries of trees proof by observing that the tree center (midpoint of diameter) is distinguished by definition Cycle-pointing (Bodirsky et al. 11): allows us to correct for symmetries of general graphs more difficult but preserves combinatorial nature of grammar (eg. can be used to build random samplers)

27 Random Sampling Boltzmann Samplers (Duchon, Flajolet, Louchard, Shaeffer 04) approximate size sampling uniform over objects of the same size allows for compositions Combinatorial class generating function Boltzmann sampler prob. drawing

28 Conclusion Summary Derived a simple, unified framework for symbolic specification of cacti Used symbolic grammar to generate large cactus graphs uniformly at random Used symbolic grammar to enumerate varieties of cactus graphs For the first time studied a graph class with a split decomposition tree that contains prime nodes Next Steps Consider other kinds of prime nodes (e.g. bipartite nodes are prime nodes for parity graphs and were studied asymptotically by Lumbroso and Shi (unpublished), but an exact enumeration is unknown)

29 Thank you!