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Composed Degree-Distance Realizations of Graphs

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Abstract

Network realization problems require, given a specification \(\pi \) for some network parameter (such as degrees, distances or connectivity), to construct a network G conforming to \(\pi \), or to determine that no such network exists. In this paper we study composed profile realization, where the given instance consists of two or more profile specifications that need to be realized simultaneously. To gain some understanding of the problem, we focus on two classical profile types, namely, degrees and distances, which were (separately) studied extensively in the past. We investigate a wide spectrum of variants of the composed distance and degree realization problem. For each variant we either give a polynomial-time realization algorithm or establish NP hardness. In particular:

  1. (i)

    We consider both precise specifications and range specifications, which specify a range of permissible values for each entry of the profile.

  2. (ii)

    We consider realizations by both weighted and unweighted graphs.

  3. (iii)

    We also study settings where the realizing graph is restricted to specific graph classes, including trees and bipartite graphs.

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Notes

  1. We consider profile types for which \(\Pi (G)\) is polynomial-time computable given G.

  2. Such a problem may arise naturally in a setting where it is known a priori that certain connections are impossible, infeasible or disallowed, due to the environment, the user specified requirements, or other reasons.

  3. For the pure degree-realization problem, there is no distinction between weighted and unweighted graphs.

  4. Note the difference between a realizable specification \(\pi \) (“\(\pi \) has a realization”) and a realizable profile type \(\Pi \) (“there is a polynomial-time algorithm deciding, for every specification \(\pi \) of \(\Pi \), if it is realizable”).

  5. Satisfying \(E\subseteq E'\).

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Acknowledgements

Supported in part by a US-Israel BSF Grant (2018043). A preliminary version was presented at IWOCA 2021.

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Authors and Affiliations

  1. City University of New York (CUNY), New York, USA

    Amotz Bar-Noy

  2. Weizmann Institute of Science, Rehovot, Israel

    David Peleg

  3. The Academic College of Tel Aviv-Yaffo, Tel Aviv, Israel

    Mor Perry

  4. Bar Ilan University, Ramat Gan, Israel

    Dror Rawitz

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  1. Amotz Bar-Noy
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  2. David Peleg
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  3. Mor Perry
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  4. Dror Rawitz
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Correspondence to Mor Perry.

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Bar-Noy, A., Peleg, D., Perry, M. et al. Composed Degree-Distance Realizations of Graphs. Algorithmica 85, 665–687 (2023). https://doi.org/10.1007/s00453-022-01055-2

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