  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.

Combinatorics