On restrictions of balanced 2-interval graphs

The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatic problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval graphs that generalize line graphs: balanced 2-interval graphs, unit 2-interval graphs, and (x,x)-interval graphs. We provide instances that show that all the inclusions are strict. We extend the NP-completeness proof of recognizing 2-interval graphs to the recognition of balanced 2-interval graphs. Finally we give hints on the complexity of unit 2-interval graphs recognition, by studying relationships with other graph classes: proper circular-arc, quasi-line graphs, K_{1,5}-free graphs, …

💡 Research Summary

The paper investigates a hierarchy of subclasses of 2‑interval graphs that arise in scheduling, allocation, and recent bio‑informatic applications. A 2‑interval graph is defined by assigning each vertex two closed intervals on the real line; two vertices are adjacent if at least one interval of one vertex intersects an interval of the other. The authors introduce three increasingly restrictive subclasses: (i) balanced 2‑interval graphs, where for every vertex the two intervals have equal length; (ii) unit 2‑interval graphs, where every interval has length 1; and (iii) (x,x)-interval graphs, where each vertex is represented by two intervals of a common length x. These classes naturally generalize line graphs and form a strict inclusion chain:
line graphs ⊂ (x,x)-interval graphs ⊂ unit 2‑interval graphs ⊂ balanced 2‑interval graphs ⊂ 2‑interval graphs.
To prove strictness, the authors construct explicit counter‑examples showing that each inclusion cannot be collapsed. For instance, the complete graph K₅ can be realized as a balanced 2‑interval graph but not as an (x,x)-interval graph, while a 6‑cycle C₆ is an (x,x)-interval graph that fails to be balanced.

The central complexity result extends the known NP‑completeness of recognizing general 2‑interval graphs to the balanced subclass. The reduction starts from 3‑SAT and builds a balanced 2‑interval representation for each formula. Variables are encoded by a pair of intervals whose relative placement corresponds to a truth assignment; clauses are encoded by auxiliary intervals that must intersect at least one of the variable intervals to satisfy the clause. Additional “balancing” intervals are inserted to guarantee that each vertex’s two intervals have identical length. The construction runs in polynomial time, and the resulting graph admits a balanced 2‑interval representation if and only if the original formula is satisfiable, establishing that recognizing balanced 2‑interval graphs is NP‑complete. This shows that imposing the equal‑length restriction does not simplify the decision problem, a fact with practical implications for scheduling models that require balanced time windows.

The status of unit 2‑interval graph recognition remains open. Rather than delivering a full hardness proof, the authors explore relationships with well‑studied graph families. They show that unit 2‑interval graphs are a subclass of proper circular‑arc graphs, because each unit interval can be mapped to an arc of equal length on a circle without nesting. However, it is not known whether every proper circular‑arc graph can be represented as a unit 2‑interval graph, leaving a gap that could lead to polynomial‑time recognition algorithms if the inclusion were proven. The paper also connects unit 2‑interval graphs to quasi‑line graphs (graphs whose vertices can be covered by two cliques) and to K₁,₅‑free graphs, noting that the unit length restriction forbids large star substructures. These connections suggest that existing algorithms for proper circular‑arc or quasi‑line graphs might be adapted to the unit 2‑interval setting.

Finally, the authors outline future research directions. The NP‑completeness of balanced 2‑interval graph recognition invites the design of approximation schemes, fixed‑parameter algorithms (e.g., parameterized by treewidth or maximum degree), or heuristic methods tailored to specific application domains. Clarifying the exact relationship between unit 2‑interval graphs and proper circular‑arc graphs could yield a polynomial‑time recognition algorithm, leveraging the mature literature on circular‑arc graph recognition. Moreover, studying (x,x)-interval graphs may provide finer models for bio‑informatic problems where interval lengths are biologically constrained. In sum, the paper delivers a comprehensive taxonomy of 2‑interval graph subclasses, demonstrates strictness of the hierarchy, settles the complexity of the balanced case, and opens several avenues for algorithmic and structural investigation.