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Criterion for a graph to admit a good orientation in terms of leaf blocks

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Abstract

Given a set S of integers, for a directed acyclic pseudograph we say that it has an S-good orientation if all its sources and sinks have degrees in S; in these terms, the existing notion of good orientation is a \(\{\,1\,\}\)-good orientation. We give a criterion for a pseudograph to admit an S-good orientation in terms of the structure of its leaf blocks. This criterion allows to compute whether a pseudograph admits an S-good orientation with optimal (linear) time and space complexity. As an application, we give a simple criterion for a graph to be realized as the Reeb graph of a Morse function in terms of leaf blocks of the graph. Similar conditions can be obtained for other classes of functions, such as (simple) Morse–Bott functions or round functions.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Code availability not applicable to this article as no code was generated or analysed during the current study.

Notes

  1. Recall that by a graph, we understand a graph allowing multiple edges and loop edges, i.e., a pseudograph. For readers who might use individual statements without reading the whole paper, we will refer to this footnote throughout the paper.

  2. The proof (to be given in a future paper) relies on the fact that the (block-cut) tree T can be oriented in such a way that its only sources and sinks be leaves, with a given non-empty set of leaves being sources and non-empty set of leaves being sinks.

  3. We thank an anonymous reviewer for suggesting this example.

  4. One may consider these estimates as expressing the complexity of the algorithm in terms of its own output. If this is undesirable, one can use (1) instead. However, there are practical situations where \(|E'|\) is known a priori to be much smaller than the bound (1). For example, consider a graph G on the set V of all people in the world, where an edge between persons A and B corresponds to sending an email message, giving a Facebook like, or having a physical contact in the context of epidemiological research. By the very nature of the modeled phenomenon we know a priori that many of multi-edges of such a graph consist of a large number of individual edges, i.e., the set \(E'\) of pairs of people having ever interacted is much smaller than the set E of their individual interactions; the number \(|E'|\) can even be estimated from sociological assumptions without actual calculation on the graph G.

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  1. CIC, Instituto Politécnico Nacional, 07338, Mexico City, Mexico

    Irina Gelbukh

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  1. Irina Gelbukh
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All authors whose names appear on the paper made substantial contributions to the conception or design of the work; or the acquisition, analysis, or interpretation of data; or the creation of new software used in the work; drafted the work or revised it critically for important intellectual content; approved the version to be published; and agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

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Communicated by Ilse Fischer.

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Gelbukh, I. Criterion for a graph to admit a good orientation in terms of leaf blocks. Monatsh Math 198, 61–77 (2022). https://doi.org/10.1007/s00605-022-01681-6

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