Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs

Bahrani, Maryam; Lumbroso, Jérémie

doi:10.1137/1.9781611975062.13

Cited by 4 publications

(4 citation statements)

References 29 publications

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“…for some γ > 0. See [13] and [2]. Moreover, the sampler for 3. in the examples can be used to sample from classes that are in bijection to these objects, for example dissections of polygons.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…As the number of non- * -vertices of R v agrees with the number d + T (v) of children of v in T , the number of edges in R v is at most O(d + T (v) 2 ). Hence the time required for transforming (T, (R v ) v∈T ) into a graph is bounded by a constant multiple of v∈T d + T (v) 2 . Hence it follows from Lemma 2.3 that for a uniformly selected n-vertex A n (T n , (R v ) v∈Tn ) the expected required time is bounded by a constant multiple of…”

Section: Main Applicationsmentioning

confidence: 99%

See 1 more Smart Citation

Exact-size Sampling of Enriched Trees in Linear Time

Παναγιώτου¹,

Ramzews²,

Stufler³

2021

Preprint

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Various combinatorial classes such as outerplanar graphs and maps, series-parallel graphs, substitution-closed classes of permutations and many more allow bijective encodings by so-called enriched trees, which are rooted trees with additional structure on the offspring of each node. Using this universal description we develop sampling procedures that uniformly generate objects from this classes with a given size n in expected time O(n). The key ingredient is a representation of enriched trees in terms of decorated Bienaymé-Galton-Watson trees, which allows us to develop a novel combination of Devroye's efficient sampler for trees [21] with Boltzmann sampling techniques. Additionally, we construct expected linear time samplers for critical Bienaymé-Galton-Watson trees having exactly n (out of ≥ n total) nodes with outdegree in some fixed set, enabling uniform generation for many combinatorial classes such as dissections of polygons.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Main Applicationsmentioning

confidence: 99%

Exact-size Sampling of Enriched Trees in Linear Time

Παναγιώτου¹,

Ramzews²,

Stufler³

2021

Preprint

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“…The classic types, such as those that appear in data structures [3,10,14,15] and digital processing [6,12,20], grow incrementally, one node at a time. In more recent times, authors considered more complex types of random graphs grown by adjoining entire graphs to a growing network [1,2,5,7,11,13,16,21]. We consider a growing network model in which the number of components attached at a stage follows a predetermined building sequence of numbers.…”

Section: Introductionmentioning

confidence: 99%

Random multi-hooking networks

Bhutani

Kalpathy

Mahmoud

2023

Prob. Eng. Inf. Sci.

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We introduce a broad class of multi-hooking networks, wherein multiple copies of a seed are hooked at each step at random locations, and the number of copies follows a predetermined building sequence of numbers. We analyze the degree profile in random multi-hooking networks by tracking two kinds of node degrees—the local average degree of a specific node over time and the global overall average degree in the graph. The former experiences phases and the latter is invariant with respect to the type of building sequence and is somewhat similar to the average degree in the initial seed. We also discuss the expected number of nodes of the smallest degree. Additionally, we study distances in the network through the lens of the average total path length, the average depth of a node, the eccentricity of a node, and the diameter of the graph.

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“…A block graph is a hooking network whose blocks are complete graphs, and a cactus graph is a hooking network whose blocks are cycles and that may include an edge K 2 in the set of blocks. The study of these graphs has been very active recently [1,2,7], with applications in genome comparison [13] as well as in telecommunication networks and material handling networks (see [4]).…”

Section: Introductionmentioning

confidence: 99%

Degree distributions of generalized hooking networks

Desmarais¹,

Holmgren²

2019

2019 Proceedings of the Sixteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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A hooking network is grown from a set of graphs called blocks, each block with a labelled vertex called a hook. At each step in the growth of the network, a vertex called a latch is chosen from the hooking network, and a block is attached by joining the hook of the block with the latch. These graphs generalize trees, which are hooking networks grown from a single edge as the only block. Using Pólya urns, we show multivariate normal limit laws for the degree distributions of hooking networks. We extend previous results by allowing for more than one block in the growth of the network and by studying arbitrarily large degrees.

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Split-Decomposition Trees with Prime Nodes: Enumeration and Random Generation of Cactus Graphs

Cited by 4 publications

References 29 publications

Exact-size Sampling of Enriched Trees in Linear Time

Exact-size Sampling of Enriched Trees in Linear Time

Random multi-hooking networks

Degree distributions of generalized hooking networks

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