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Axiomatizing Rectangular Grids with no Extra Non-unary Relations

Kopczyński Eryk

Fundamenta Informaticae

2020

Vol. 176, nr 2

129--138

We construct a first-order formula φ such that all finite models of φ are non-narrow rectangular grids without using any binary relations other than the grid neighborship relations. As a corollary, we prove that a set A ⊆ ℕ is a spectrum of a formula which has only planar models if numbers n ∈ A can be recognized by a non-deterministic Turing machine (or a one-dimensional cellular automaton) in time t (n) and space s (n), where t (n)s (n) ≤ n and t (n), s (n) = Ω(log(n)).

Packing two copies of a tree into a planar graph

Woźniak M.

Opuscula Mathematica

2003

Vol. 23

95-97

A simple proof is given for a result of Garcia, Hernando, Hurtado, Noy and Tehel on packing of two copies of a tree into a planar graph.