Irreducible pairings and indecomposable tournaments

We only consider finite structures. With every totally ordered set $V$ and a subset $P$ of $\binom{V}{2}$, we associate the underlying tournament ${\rm Inv}(\underline{V}, P)$ obtained from the transitive tournament $\underline{V}:=(V, \{(x,y) \in V \times V : x < y \})$ by reversing $P$, i.e., by reversing the arcs $(x,y)$ such that $\{x,y\} \in P$. The subset $P$ is a pairing (of $\cup P$) if $|\cup P| = 2|P|$, a quasi-pairing (of $\cup P$) if $|\cup P| = 2|P|-1$; it is irreducible if no nontrivial interval of $\cup P$ is a union of connected components of the graph $(\cup P, P)$. In this paper, we consider pairings and quasi-pairings in relation to tournaments. We establish close relationships between irreducibility of pairings (or quasi-pairings) and indecomposability of their underlying tournaments under modular decomposition. For example, given a pairing $P$ of a totally ordered set $V$ of size at least $6$, the pairing $P$ is irreducible if and only if the tournament ${\rm Inv}(\underline{V}, P)$ is indecomposable. This is a consequence of a more general result characterizing indecomposable tournaments obtained from transitive tournaments by reversing pairings. We obtain analogous results in the case of quasi-pairings.Comment: 17 page

Graphs whose indecomposability graph is 2-covered

Given a graph $G=(V,E)$, a subset $X$ of $V$ is an interval of $G$ provided that for any $a, b\in X$ and $x\in V \setminus X$, $\{a,x\}\in E$ if and only if $\{b,x\}\in E$. For example, $\emptyset$, $\{x\}(x\in V)$ and $V$ are intervals of $G$, called trivial intervals. A graph whose intervals are trivial is indecomposable; otherwise, it is decomposable. According to Ille, the indecomposability graph of an undirected indecomposable graph $G$ is the graph $\mathbb I(G)$ whose vertices are those of $G$ and edges are the unordered pairs of distinct vertices $\{x,y\}$ such that the induced subgraph $G[V \setminus \{x,y\}]$ is indecomposable. We characterize the indecomposable graphs $G$ whose $\mathbb I(G)$ admits a vertex cover of size 2.Comment: 31 pages, 5 figure

Modular Decomposition and the Reconstruction Conjecture

We prove that a large family of graphs which are decomposable with respect to the modular decomposition can be reconstructed from their collection of vertex-deleted subgraphs.Comment: 9 pages, 2 figure

Tame Decompositions and Collisions

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we have reasonable bounds on the number of decomposables of degree n. Nevertheless, no exact formula is known if $n$ has more than two prime factors. In order to count the decomposables, one wants to know, under a suitable normalization, the number of collisions, where essentially different (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all 2-collisions. We introduce a normal form for multi-collisions of decompositions of arbitrary length with exact description of the (non)uniqueness of the parameters. We obtain an efficiently computable formula for the exact number of such collisions at degree n over a finite field of characteristic coprime to p. This leads to an algorithm for the exact number of decomposable polynomials at degree n over a finite field Fq in the tame case

Convex circuit free coloration of an oriented graph

We introduce the \textit{convex circuit-free coloration} and \textit{convex circuit-free chromatic number} $\overrightarrow{\chi_a}(\overrightarrow{G})$ of an oriented graph $\overrightarrow{G}$ and establish various basic results. We show that the problem of deciding if an oriented graph verifies $\chi_a( \overrightarrow{G}) \leq k$ is NP-complete if $k \geq 3$ and polynomial if $k \leq 2$. We exhibit an algorithm which finds the optimal convex circuit-free coloration for tournaments, and characterize the tournaments that are \textit{vertex-critical} for the convex circuit-free coloration