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Characterization of B0-VPG Cocomparability Graphs and a 2D Visualization of their Posets

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Abstract

B0-VPG graphs are intersection graphs of axis-parallel line segments in the plane. Cohen et al. (Order 33(1), 39–49, 2016) pose the question of characterizing B0-VPG permutation graphs. We respond here by characterizing B0-VPG cocomparability graphs. This helps us show that a simple necessary condition in fact characterizes B0-VPG permutation graphs. The characterization also leads to a polynomial time recognition algorithm and its proof gives us a B0-VPG drawing algorithm for the class of B0-VPG cocomparability graphs. Our drawing algorithm starts by fixing any one of the many posets P whose cocomparability graph is the input graph G. On the set of axis-parallel line segments in the plane, we define a binary relation “≺2” as p ≺2q if and only if they are non-intersecting and the bottom-left endpoint of p precedes the top-right endpoint of q in the product order on \(\mathbb {R}^{2}\). The reflexive closure ≼2 of the relation ≺2 restricted to the line segments of our drawing turns out to be a partial order isomorphic to the poset P.

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Acknowledgements

We thank the anonymous reviewers for their comments and suggestions which have made this paper easier to follow. The proofs of the Lemmas 2 and 4 in particular have improved considerably.

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Authors and Affiliations

  1. Indian Institute of Technology Palakkad, Palakkad, India

    Sreejith K. Pallathumadam & Deepak Rajendraprasad

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  1. Sreejith K. Pallathumadam
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  2. Deepak Rajendraprasad
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Correspondence to Sreejith K. Pallathumadam.

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Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

An earlier version of these results were presented in the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020) [27]. This new version contains the full proofs of the characterization result and a new visual interpretation of the order relation in the resulting drawing.

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

Appendix: Additional Figures

Appendix: Additional Figures

In this section, we include some additional figures which will help in understanding our proofs better. Fig. 6 corresponds Lemma 4.(3.v). Figures 7, 8 and 9 correspond to the proof in Section 3.1.

Fig. 6
figure 6

Minimal graph RM having a C3 in which each edge is part of an induced C4. For i ∈{0,1,2}, branch sets \(B_{i^{\prime }}\) and \(B_{i^{\prime \prime }}\) may not be distinct

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Fig. 7
figure 7

The only valid configurations when x and y are in adjacent branch sets, x ≺σy and Ix contains the intersection point

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Fig. 8
figure 8

The only valid configurations when x and y are in adjacent branch sets, x ≺σy and neither Ix nor Iy contains the intersection point

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Fig. 9
figure 9

\(\mathcal {I}_{i}\) and \(\mathcal {I}_{j}\) are of the same parity and Iy is to the top-left of Ix

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Pallathumadam, S.K., Rajendraprasad, D. Characterization of B0-VPG Cocomparability Graphs and a 2D Visualization of their Posets. Order 39, 465–484 (2022). https://doi.org/10.1007/s11083-021-09589-w

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