Abstract
A path is a simple, piecewise linear curve made up of alternating horizontal and vertical line segments in the plane. A k-bend path is a path made up of at most k + 1 line segments. A Bk-VPG representation of a graph is a collection of k-bend paths such that each path in the collection represents a vertex of the graph and two such paths intersect if and only if the vertices they represent are adjacent in the graph. The graphs that have a Bk-VPG representation are calledBk-VPG graphs. The bend number of a graph G, denoted by bend(G), is the minimum integer k for which G has a Bk-VPG representation. In this paper, we study the relationship between poset dimension and bend number of cocomparability graphs. It is known that the poset dimension dim(G) of a cocomparability graph G is greater than or equal to its bend number bend(G). Cohen et al. (order2016) asked for examples of cocomparability graphs with low bend number and high poset dimension. We answer this question by proving that for each \(m, t \in \mathbb {N}\), there exists a cocomparability graph Gt, m with t < bend(Gt, m) ≤ 4t + 29 and dim(Gt, m) − bend(Gt, m) > m. The techniques used to prove the above result allow us to partially address the open question posed by Chaplick et al. (wg2012) who asked whether Bk-VPG-chordal \(\subsetneq ~B_{k + 1}\)-VPG-chordal for all \(k \in \mathbb {N}\). We address this by proving that there are infinitely many \(m \in \mathbb {N}\) such that Bm-VPG-split \(\subsetneq ~B_{m + 1}\)-VPG-split which provides infinitely many positive examples. We use the same techniques to prove that, for all \(t \in \mathbb {N}\), Bt-VPG-\(Forb(C_{\geq 5})~\subsetneq ~B_{4t + 29}\)-VPG-Forb(C≥ 5), where Forb(C≥ 5) denotes the family of graphs that does not contain induced cycles of length greater than 4. Furthermore, we show that for all \(t \in \mathbb {N}\), PBt-VPG-split \(\subsetneq PB_{36t + 80}\)-VPG-split, where PBt-VPG denotes the class of graphs with proper bend number at most t (i.e. it has a Bt-VPG representation in which two paths have only finitely many intersection points, each intersection point belongs to exactly two paths, and whenever two paths intersect they cross each other).
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Acknowledgements
We thank Sagnik Sen for helpful comments in preparing the manuscript. We thank Subhodeep Ranjan Dev for carefully reading the final draft. We would like to thank the anonymous referees for meticulously reading the manuscript, and for helpful suggestions which made the proofs more rigorous and increased the readability and flow of the manuscript. Joydeep Mukherjee is supported by DST SERB NPDF fellowship (PDF/2016/001647).
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Chakraborty, D., Das, S., Mukherjee, J. et al. Bounds on the Bend Number of Split and Cocomparability Graphs. Theory Comput Syst 63, 1336–1357 (2019). https://doi.org/10.1007/s00224-019-09912-4
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DOI: https://doi.org/10.1007/s00224-019-09912-4