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

Description of the Tournaments Which are Reconstructible from Their k-cycle Partial Digraphs for k∈{3,4}k \in \{3, 4\} k ∈ { 3 , 4 }

International audienceIf T is a tournament on n vertices and k is an integer with , then the k-cycle partial digraph of T, denoted by T(k), is the spanning subdigraph of T for which the arcs are those of the k-cycles of T. In 1989, Thomassen proved that given two irreducible tournaments T and on the same vertex set with at least 6 vertices, if , then . This result allows us to introduce the following notion of reconstruction. A tournament T is reconstructible from its k-cycle partial digraph whenever is isomorphic to T for each tournament such that is isomorphic to T(k). In this paper, we give a complete description of the tournaments which are reconstructible from their k-cycle partial digraphs where . For , our proof is based on the above result of Thomassen. For , we introduce and study the tournaments for which all modules are irreducible. We use properties of the dilatation operator to describe such tournaments, and then we obtain a new modular decomposition of tournaments sometimes finer than that of Gallai