The combinatorics of splittability

Splittings and Ramsey Properties of Permutation Classes

We say that a permutation π is merged from permutations ρ and τ , if we can color the elements of π red and blue so that the red elements are order-isomorphic to ρ and the blue ones to τ . A permutation class is a set of permutations closed under taking subpermutations. A permutation class C is splittable if it has two proper subclasses A and B such that every element of C can be obtained by me...

Degree-Splittability of Multigraphs and Caterpillars

A multigraph is degree-splittable if it decomposes into two submultigraphs whose degree lists are the same, plus one leftover edge when the total number of edges is odd. We prove that a connected multigraph is degree-splittable if it has an even number of vertices of each odd degree. For caterpillars, we characterize the splittable caterpillars with diameter at most 4, provide a general suffici...

Splittability and Low Separation Axioms

The power and usefulness of cleavability (also known as splittability) have been well established within the framework of topology by A.V. Arhangel’skii and his associates. The concept was transferred to partially ordered sets by D.J. Marron and T.B.M. McMaster. The connections between topology and order are here exploited as we examine the properties enjoyed by a topological space that is spli...

Splittability of Bilexical Context-Free Grammars is Undecidable

Bilexical context-free grammars (2-LCFGs) have proved to be accurate models for statistical natural language parsing. Existing dynamic programming algorithms used to parse sentences under these models have running time of O(|w|4), where w is the input string. A 2-LCFG is splittable if the left arguments of a lexical head are always independent of the right arguments, and vice versa. When a 2-LC...

Billiard Sequences and the Property of Splittability of Integrable Hamilton Systems