On characterization of periodic digraphs

Pavlenko, V. A.

doi:10.1007/bf01074883

Cited by 4 publications

(6 citation statements)

References 2 publications

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“…Therefore, every digraph is a subgraph of an M-graph. Moreover, as Pavlenko showed in [8] for any n ≥ 1 there exists m ≥ 2n − 1 and m-vertex periodic digraph which contain K n as a subgraph.…”

Section: Transformations Of M-graphsmentioning

confidence: 81%

“…The first graph-theoretic results concerning periodic graphs were obtained by Pavlenko [6,7,8]. In particular, in [6] the number of non-isomorphic periodic graphs with a given number of vertices was counted.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, in [6] the number of non-isomorphic periodic graphs with a given number of vertices was counted. Criteria for an arbitrary digraph to be a periodic graph or to be an induced subgraph of a periodic graph were presented in [7] and [8], respectively.…”

Section: Introductionmentioning

confidence: 99%

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On the Abstract Properties of Markov Graphs for Maps on Trees

Kozerenko¹

2017

matbil

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Having a dynamical system on the vertex set of a finite tree, one can construct the corresponding Markov graph which is the digraph that encodes covering relation between edges in a tree. Representatives of isomorphism classes of Markov graphs are called M-graphs. In this paper we prove that the class of M-graphs is closed under several prescribed digraph transformations (such as deletion of a vertex in a digraph or taking the disjoint union of digraphs, for example). We also give a complete list of tournaments which are M-graphs as well as of M-graphs with three vertices.

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Section: Transformations Of M-graphsmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

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On the Abstract Properties of Markov Graphs for Maps on Trees

Kozerenko¹

2017

matbil

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“…Such a crucial role that Markov graphs play in combinatorial dynamics is a reason to study these digraphs from graph-theoretic point of view. It seems that the first results in this direction were obtained by Pavlenko [7,8,9]. In particular, the number of non-isomorphic periodic graphs with given number of vertices was calculated in [7].…”

Section: Introductionmentioning

confidence: 99%

Linear and metric maps on trees via Markov graphs

Kozerenko¹

2018

Commentationes Mathematicae Universitatis Carolinae

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The main focus of combinatorial dynamics is put on the structure of periodic points (and the corresponding orbits) of topological dynamical systems. The first result in this area is the famous Sharkovsky's theorem which completely describes the coexistence of periods of periodic points for a continuous map from the closed unit interval to itself. One feature of this theorem is that it can be proved using digraphs of a special type (the so-called periodic graphs). In this paper we use Markov graphs (which are the natural generalization of periodic graphs in case of dynamical systems on trees) as a tool to study several classes of maps on trees. The emphasis is put on linear and metric maps.

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“…The graph-theoretic criterion for periodic graphs was obtained in [8]. In [5] the pairs (X, σ) were characterized for several prescribed classes of Markov graphs Γ(X, σ).…”

Section: Introductionmentioning

confidence: 99%

On expansive and anti-expansive tree maps

Kozerenko

2018

OpMath

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Abstract. With every self-map on the vertex set of a finite tree one can associate the directed graph of a special type which is called the Markov graph. Expansive and anti-expansive tree maps are two extremal classes of maps with respect to the number of loops in their Markov graphs. In this paper we prove that a tree with at least two vertices has a perfect matching if and only if it admits an expansive cyclic permutation of its vertices. Also, we show that for every tree with at least three vertices there exists an expansive map with a weakly connected (strongly connected provided the tree has a perfect matching) Markov graph as well as anti-expansive map with a strongly connected Markov graph.

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On characterization of periodic digraphs

Cited by 4 publications

References 2 publications

On the Abstract Properties of Markov Graphs for Maps on Trees

On the Abstract Properties of Markov Graphs for Maps on Trees

Linear and metric maps on trees via Markov graphs

On expansive and anti-expansive tree maps

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