Abstract
A collection of linear orders on X, say \({\mathscr{L}}\), is said to realize a partially ordered set (or poset) \(\mathcal {P} = (X, \preceq )\) if, for any two distinct x,y ∈ X, x ≼ y if and only if x ≺Ly, \(\forall L \in {\mathscr{L}}\). We call \({\mathscr{L}}\) a realizer of \(\mathcal {P}\). The dimension of \(\mathcal {P}\), denoted by \(dim(\mathcal {P})\), is the minimum cardinality of a realizer of \(\mathcal {P}\). A containment model \(M_{\mathcal {P}}\) of a poset \(\mathcal {P}=(X,\preceq )\) maps every x ∈ X to a set Mx such that, for every distinct x,y ∈ X, x ≼ y if and only if \(M_{x} \varsubsetneq M_{y}\). We shall be using the collection (Mx)x∈X to identify the containment model \(M_{\mathcal {P}}\). A poset \(\mathcal {P}=(X,\preceq )\) is a Containment order of Paths in a Tree (CPT poset), if it admits a containment model \(M_{\mathcal {P}}=(P_{x})_{x \in X}\) where every Px is a path of a tree T, which is called the host tree of the model. We show that if a poset \(\mathcal {P}\) admits a CPT model in a host tree T of maximum degree Δ and radius r, then \(dim(\mathcal {P}) \leq \lg \lg {\Delta } + (\frac {1}{2} + o(1))\lg \lg \lg {\Delta } + \lg r + \frac {1}{2} \lg \lg r + \frac {1}{2}\lg \pi + 3\). This bound is asymptotically tight up to an additive factor of \(\min \limits (\frac {1}{2}\lg \lg \lg {\Delta }, \frac {1}{2}\lg \lg r)\). Further, let \(\mathcal {P}(1,2;n)\) be the poset consisting of all the 1-element and 2-element subsets of [n] under ‘containment’ relation and let dim(1,2;n) denote its dimension. The proof of our main theorem gives a simple algorithm to construct a realizer for \(\mathcal {P}(1,2;n)\) whose cardinality is only an additive factor of at most \(\frac {3}{2}\) away from the optimum.
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Acknowledgements
The authors are grateful to Prof. William T. Trotter for his detailed review comments that helped improve this manuscript significantly. We also thank Prof. Martin Golumbic and Prof. Vincent Limouzy for fruitful discussions on the topic of CPT graphs and posets.
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Majumder, A., Mathew, R. & Rajendraprasad, D. Dimension of CPT Posets. Order 38, 13–19 (2021). https://doi.org/10.1007/s11083-020-09524-5
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DOI: https://doi.org/10.1007/s11083-020-09524-5