Letter graphs and geometric grid classes of permutations: Characterization and recognition

Abstract

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and k-letter graphs for a fixed k. However, constructive algorithms are available only for k = 2. In this paper, we present the first constructive polynomial-time algorithm for the recognition of 3-letter graphs. It is based on a structural characterization of graphs in this class. * An extended abstract of this paper was published in [2].

Letter Graphs and Geometric Grid Classes of Permutations: Characterization and Recognition

“…Note that in the proof of this result, we are able to use the same word to encode both the permutation π ∈ Geom(M ) and the letter graph satisfying Γ D (w) ∼ = G π . Theorem 5.5 (Alecu, Lozin, de Werra, and Zamaraev [6,7]). If the permutation class C is geometrically griddable, then the corresponding graph class G C has bounded lettericity.…”

“…Instead of trying to choose one geometric gridding for each member of a geometric grid class, we seek only to show that certain griddings are geometric. Returning to this goal, the easier direction of Theorem 1.1-which states that the graph class associated to a geometrically griddable permutation class has bounded lettericity-has already been proved in [6,7], but we include another proof here, both for completeness and because it further illustrates some ideas used in the proof of the converse. Note that in the proof of this result, we are able to use the same word to encode both the permutation π ∈ Geom(M ) and the letter graph satisfying Γ D (w) ∼ = G π .…”

Understanding lettericity I: a structural hierarchy

“…In [3] and [2], we showed that a permutation class X is geometrically griddable if and only if the corresponding class G X of permutation graphs has bounded lettericity. The "only if" direction of the statement is fairly straightforward: for a permutation π with a geometric gridding by a partial multiplication matrix M , one can carefully construct a decoder D over the "cell alphabet" of M (whose letters are the non-empty cells of the M ).…”

“…A word w such that G(D, w) ∼ = G π is then given by the encoding ϕ from Subsection 2.3. The "if" direction, shown in [2], proved to be considerably more tricky from a technical perspective. At the heart of the argument, however, is a simple idea: one first uses Theorem 5 to deduce that X is monotone griddable.…”

“…This notion also attracted considerable attention from researchers in the field. However, it was not until 2020 that a close relationship between the two notionsletter graphs and geometric griddability -was discovered and described, first, in one direction [3], and then in the other [2]. Informally, this relationship can be characterised as follows: graph lettericity, restricted to the class of permutation graphs, and geometric griddability describe the same concept in the language of graphs and permutations, respectively.…”