Circle graphs and monadic second-order logic

Abstract

This article is part of a project consisting in expressing, whenever possible, graph properties and graph transformations in monadic second-order logic or in its extensions using modulo p cardinality set predicates or auxiliary linear orders. A circle graph is the intersection graph of a set of chords of a circle. Such a set is called a chord diagram. It can also be described by a word with two occurrences of each letter, called a double occurrence word. If a circle graph is prime for the split (or join) decomposition defined by Cunnigham, it has a unique representation by a chord diagram, and this diagram can be defined by monadic secondorder formulas with the even cardinality set predicate. By using the (canonical) split decomposition of a circle graph, we define in monadic second-order logic with auxiliary linear orders all its chord diagrams. This construction uses the fact that the canonical split decomposition of a graph can be constructed in monadic second-order logic with help of an arbitrary linear order. We prove that the order of first occurrences of the letters in a double occurrence word w that represents a connected circle graph determines this word in a unique way. The word w can be defined by a monadic second-order formula from the word of first occurrences of letters. We also prove that a set of circle graphs has bounded clique-width if and only if all the associated chord diagrams have bounded tree-width.

Extending partial representations of circle graphs

“…Our result can be interpreted as "describing a structure like PQ-trees 1 for circle F I G U R E 2 A self-intersecting closed curve with n intersections numbered n 1,…, corresponds to a representation of circle graph with the vertices n 1,…, where the endpoints of the chords are placed according to graphs." It is possible that the proof techniques from other papers on circle graphs such as [13,23] would give a similar description. However, these techniques are more involved than our approach which turns out to be quite elementary and simple.…”

“…In Section , we give a simple recursive description of all possible representations based on splits. Our result can be interpreted as “describing a structure like PQ‐trees for circle graphs.” It is possible that the proof techniques from other papers on circle graphs such as would give a similar description. However, these techniques are more involved than our approach which turns out to be quite elementary and simple.…”

Mutant knots and intersection graphs

“…This theorem is contained implicitly in papers [3,8,11] where chord diagrams are written as double occurrence words, the language better suitable for describing algorithms than for topological explanation.…”

Insertions Yielding Equivalent Double Occurrence Words

“…Cruz, M.M. Ferrari, N. Jonoska, L. Nabergall, M. Saito / Insertions on Double Occurrence Words [8], and algebraic combinatorics [17]. DOWs are also known as Gauss words and are closely related to linear diagrams, chord diagrams, and circle graphs.…”

“…We use standard definitions and conventions (e.g., [7,8,13,11]). A word w over Σ is a finite sequence of symbols a 1 · · · a n in Σ; the length of w, denoted |w|, is n. The set of all words over Σ is denoted by Σ * and includes the empty word whose length is 0; and Σ + = Σ * \ { }.…”