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The  graph of irreducible parabolic subgroups  is a combinatorial object associated to an Artin–Tits group   A   defined so as to coincide with the curve graph of the    (  n  +  1  )   ‐times punctured disk when   A   is Artin's braid group on    (  n  +  1  )    strands. In this case, it is a hyperbolic graph, by the celebrated Masur–Minsky's theorem. Hyperbolicity of the graph of irreducible parabolic subgroups for more general Artin–Tits groups is an important open question. In this paper, we give a partial affirmative answer.  For    n  ⩾  3   , we show that the graph of irreducible parabolic subgroups associated to the Artin–Tits group of spherical type    B  n    is also isomorphic to the curve graph of the    (  n  +  1  )   ‐times punctured disk; hence, it is hyperbolic.  For    n  ⩾  2   , we show that the graphs of irreducible parabolic subgroups associated to the Artin–Tits groups of euclidean typeA  ∼   n    andC  ∼   n    are isomorphic to some subgraphs of the curve graph of the    (  n  +  2  )   ‐times punctured disk which are not quasi‐isometrically embedded. We prove nonetheless that these graphs are hyperbolic.

Curve graphs for Artin–Tits groups of type B , A ∼ and C ∼ are hyperbolic