Counting vertices and cubes in median graphs of circular split systems

Choe, Youngbin; Huber, Katharina T.; Koolen, Jack H.; Kwon, Young Soo; Moulton, Vincent

doi:10.1016/j.ejc.2007.02.003

Cited by 5 publications

(1 citation statement)

References 16 publications

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“…The structure of median graphs is well understood, see the survey [10], recent papers [4][5][6]13], and references therein. Most of the numerous studies of these graphs focus on finite graphs, nonetheless, [1,2,14] are important references for the infinite case.…”

Section: Introductionmentioning

confidence: 99%

Transitive, Locally Finite Median Graphs with Finite Blocks

Imrich

Klavžar

2009

Graphs and Combinatorics

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The subject of this paper are infinite, locally finite, vertex-transitive median graphs. It is shown that the finiteness of the -classes of such graphs does not guarantee finite blocks. Blocks become finite if, in addition, no finite sequence of -contractions produces new cutvertices. It is proved that there are only finitely many vertex-transitive median graphs of given finite degree with finite blocks. An infinite family of vertex-transitive median graphs with finite intransitive blocks is also constructed and the list of vertex-transitive median graphs of degree four is presented.

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Section: Introductionmentioning

confidence: 99%

Transitive, Locally Finite Median Graphs with Finite Blocks

Imrich

Klavžar

2009

Graphs and Combinatorics

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Complexity of Phylogenetic Networks: Counting Cubes in Median Graphs and Related Problems

Kovše

2009

Analysis of Complex Networks

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Injective Split Systems

Hellmuth

Huber

Moulton

et al. 2023

Graphs and Combinatorics

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A split system $$\mathcal S$$ S on a finite set X, $$|X|\ge 3$$ | X | ≥ 3 , is a set of bipartitions or splits of X which contains all splits of the form $$\{x,X-\{x\}\}$$ { x , X - { x } } , $$x \in X$$ x ∈ X . To any such split system $$\mathcal S$$ S we can associate the Buneman graph $$\mathcal B(\mathcal S)$$ B ( S ) which is essentially a median graph with leaf-set X that displays the splits in $$\mathcal S$$ S . In this paper, we consider properties of injective split systems, that is, split systems $$\mathcal S$$ S with the property that $${{\,\textrm{med}\,}}_{\mathcal B(\mathcal S)}(Y) \ne {{\,\textrm{med}\,}}_{\mathcal B(\mathcal S)}(Y')$$ med B ( S ) ( Y ) ≠ med B ( S ) ( Y ′ ) for any 3-subsets $$Y,Y'$$ Y , Y ′ in X, where $${{\,\textrm{med}\,}}_{\mathcal B(\mathcal S)}(Y)$$ med B ( S ) ( Y ) denotes the median in $$\mathcal B(\mathcal S)$$ B ( S ) of the three elements in Y considered as leaves in $$\mathcal B(\mathcal S)$$ B ( S ) . In particular, we show that for any set X there always exists an injective split system on X, and we also give a characterization for when a split system is injective. We also consider how complex the Buneman graph $$\mathcal B(\mathcal S)$$ B ( S ) needs to become in order for a split system $$\mathcal S$$ S on X to be injective. We do this by introducing a quantity for |X| which we call the injective dimension for |X|, as well as two related quantities, called the injective 2-split and the rooted-injective dimension. We derive some upper and lower bounds for all three of these dimensions and also prove that some of these bounds are tight. An underlying motivation for studying injective split systems is that they can be used to obtain a natural generalization of symbolic tree maps. An important consequence of our results is that any three-way symbolic map on X can be represented using Buneman graphs.

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Counting vertices and cubes in median graphs of circular split systems

Cited by 5 publications

References 16 publications

Transitive, Locally Finite Median Graphs with Finite Blocks

Transitive, Locally Finite Median Graphs with Finite Blocks

Complexity of Phylogenetic Networks: Counting Cubes in Median Graphs and Related Problems

Injective Split Systems

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