Relationships between 2-interval graphs and balanced/disjoint balanced 3-interval graphs

Determine the inclusion or comparability relationships between the class of 2-interval graphs and the classes of balanced 3-interval graphs and disjoint balanced 3-interval graphs, clarifying whether any of these classes contain one another or are incomparable.

Background

In summarizing the landscape of subclasses, the authors indicate several proven containments but explicitly note that the relation between 2-interval graphs and balanced/disjoint balanced 3-interval graphs remains unresolved.

Establishing these relationships would complete the structural picture connecting 2-interval and 3-interval subclasses, with implications for representation constraints and algorithmic properties.