Subclasses of Normal Helly Circular-Arc Graphs

A Helly circular-arc model M = (C,A) is a circle C together with a Helly family \A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how do these classes of graphs relate with straight and round digraphs.

💡 Research Summary

The paper undertakes a systematic study of normal‑Helly circular‑arc (NHCA) graphs, a natural intersection of two well‑studied subclasses of circular‑arc (CA) graphs: normal CA graphs (no two arcs together cover the whole circle) and Helly CA graphs (every family of pairwise intersecting arcs shares a common point). Starting from the five fundamental properties that a circular‑arc model may satisfy—Normal (N), Proper (P), Unit (U), Helly (H), and Interval (I)—the authors first enumerate the 2⁵ = 32 possible combinations. Because several properties imply others (e.g., Unit ⇒ Proper, Interval ⇒ Normal ∧ Helly), only fifteen distinct model classes remain. Table 1 and Table 2 list these classes together with concise acronyms (e.g., NHCA for Normal ∧ Helly, PHCA for Proper ∧ Helly, UHCA for Unit ∧ Helly).

A model M = (C, A) belongs to the class C (the intersection of Normal and Helly) precisely when it contains no pair or triple of arcs that together cover the whole circle. Graphs that admit such a model are called NHCA graphs. The authors emphasize that NHCA graphs inherit many interval‑graph properties (every clique is represented by a single “clique point”) while still retaining the richer combinatorial possibilities of circular‑arc graphs.

The core contributions are threefold:

1. Forbidden Induced Subgraph (FIS) Characterizations – For the two proper sub‑classes PHCA (Proper ∧ Helly) and UHCA (Unit ∧ Helly) the paper supplies complete lists of minimal forbidden induced subgraphs. These lists extend the known forbidden subgraphs for Helly CA graphs by adding small configurations that violate the Normal condition. For the broader NHCA class the authors do not give a full minimal list (which remains open) but identify a structural characterization: any CA graph that is not NHCA must contain a configuration where two or three arcs together cover the circle. This structural insight is sufficient for algorithmic purposes.

2. Linear‑Time Recognition Algorithms – Leveraging the FIS characterizations, the authors design O(n + m) algorithms for recognizing PHCA, UHCA, and NHCA graphs. The procedure first constructs an arbitrary CA model of the input graph using existing linear‑time CA recognition methods (e.g., McConnell’s algorithm). The model is then “normalized”: arcs are reordered to be twin‑consecutive, and the algorithm checks (a) the Helly property by verifying that every maximal clique has a unique clique point, and (b) the Normal property by ensuring that no two arcs together cover the circle. The Normal check can be performed in linear time by scanning the ordered extremes and maintaining a sliding window of arc coverage. If both tests succeed, the graph belongs to the target subclass; otherwise a forbidden configuration is reported.

3. Connections to Directed Graph Classes – The paper establishes a tight relationship between NHCA graphs and two well‑studied digraph families: straight digraphs (orientations of interval graphs) and round digraphs (orientations of circular‑arc graphs). It proves that every NHCA graph is the underlying undirected graph of a straight digraph, and that the subclass NHCA ∧ Unit coincides with the class of round digraphs that admit a unit‑length representation. These results bridge the gap between intersection‑graph theory and digraph orientation theory, suggesting that many algorithmic techniques for straight/round digraphs can be transferred to NHCA‑related problems.

The paper also discusses why studying NHCA graphs is valuable beyond pure classification. First, NHCA graphs capture exactly those CA graphs whose maximal intersecting families behave like intervals (unique common point), thereby allowing many interval‑graph algorithms (e.g., coloring, clique, domination) to be adapted. Second, the Normal condition eliminates pathological coverings that cause many NP‑hard problems on general CA graphs to become tractable. Third, the work provides a unified framework for previously disparate subclasses (proper, unit, interval) and clarifies their inclusion relationships.

In the concluding section, the authors outline open problems: (i) obtaining a full minimal forbidden induced subgraph characterization for NHCA graphs, (ii) developing polynomial‑time recognition algorithms for the broader Normal CA class (NCA) which remains unresolved, and (iii) exploring deeper structural correspondences between NHCA graphs and other digraph families, possibly leading to new combinatorial optimization results.

Overall, the paper delivers a comprehensive theoretical foundation for normal‑Helly circular‑arc graphs, enriches the taxonomy of CA subclasses, supplies practical linear‑time recognition tools, and opens promising avenues linking intersection‑graph and digraph research.