Obstructions to chordal circular-arc graphs of small independence number

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerker and Boland. We show that a circular-arc graph cannot have a blocking quadruple. We also observe that the absence of blocking quadruples is not in general sufficient to guarantee that a graph is a circular-arc graph. Nonetheless, it can be shown to be sufficient for some special classes of graphs, such as those investigated by Bonomo et al. In this note, we focus on chordal graphs, and study the relationship between the structure of chordal graphs and the presence/absence of blocking quadruples. Our contribution is two-fold. Firstly, we provide a forbidden induced subgraph characterization of chordal graphs without blocking quadruples. In particular, we observe that all the forbidden subgraphs are variants of the subgraphs forbidden for interval graphs. Secondly, we show that the absence of blocking quadruples is sufficient to guarantee that a chordal graph with no independent set of size five is a circular-arc graph. In our proof we use a novel geometric approach, constructing a circular-arc representation by traversing around a carefully chosen clique tree.

💡 Research Summary

The paper introduces a novel structural concept called a blocking quadruple (BQ) and investigates its role in the relationship between chordal graphs and circular‑arc graphs. A BQ consists of four vertices {a, b, c, d} such that for any pair among them, either the two vertices completely miss every path connecting the remaining two vertices, or they are connected by a path that is entirely missed by the other two vertices. This definition generalizes the well‑known asteroidal triple (AT) used in the classical Lekkerkerker–Boland characterization of interval graphs, extending the “path‑blocking” idea from three to four vertices.

The authors first prove that no circular‑arc graph can contain a BQ. In a circular‑arc representation each vertex corresponds to an arc on a circle, and adjacency is expressed by overlapping arcs. The BQ conditions would require two arcs to either avoid or completely block the arcs representing the other two vertices, which is geometrically impossible on a circle. Consequently, the presence of a BQ is a forbidden configuration for circular‑arc graphs.

However, the absence of a BQ does not guarantee that a graph is a circular‑arc graph. The paper supplies a concrete counterexample: a chordal graph that contains no BQ yet fails to admit any circular‑arc representation. This shows that BQ‑freeness is a necessary but not sufficient condition in general.

The main contributions are twofold. First, the authors provide a complete forbidden‑induced‑subgraph characterization of chordal graphs that are BQ‑free. All minimal forbidden subgraphs turn out to be variants of the classic forbidden structures for interval graphs (e.g., induced 4‑cycles, graphs containing an asteroidal triple). By proving that any chordal graph containing one of these subgraphs must also contain a BQ, they establish that BQ‑free chordal graphs are exactly those that avoid this specific list.

Second, they prove a sufficiency result for a restricted subclass of chordal graphs: if a chordal graph has independence number α(G) ≤ 4 (i.e., it contains no independent set of size five) and is BQ‑free, then it is necessarily a circular‑arc graph. The proof employs a novel geometric construction based on a carefully selected clique tree. The steps are as follows:

1. Clique‑tree selection – Every chordal graph admits a tree whose nodes are its maximal cliques and whose edges represent non‑empty intersections. The authors choose a root and order the cliques by a preorder traversal.

2. Arc assignment – Each maximal clique is mapped to a contiguous interval (arc) on a circle. The length of the interval reflects the size of the clique, and overlapping intervals correspond to the intersection of adjacent cliques in the tree.

3. Traversal and stitching – As the traversal proceeds, the algorithm places the next clique’s arc so that it overlaps precisely with the arc of its parent clique on the set of shared vertices. This guarantees that any two vertices belonging to a common maximal clique receive overlapping arcs, while vertices that never share a clique receive disjoint arcs.

4. Role of the independence bound – If α(G) ≥ 5, at some stage five mutually non‑adjacent vertices would have to be placed on the circle without any pair of arcs overlapping, which forces a conflict with the required overlaps dictated by the tree structure. When α(G) ≤ 4, the “space” on the circle is sufficient to accommodate all arcs without violating the overlap constraints, and the BQ‑free condition ensures that no hidden obstruction appears.

The resulting set of arcs yields a valid circular‑arc representation, establishing that every BQ‑free chordal graph with independence number at most four belongs to the circular‑arc class.

In summary, the paper identifies BQ as a natural four‑vertex analogue of the asteroidal triple, shows that BQ‑freeness characterizes a subclass of chordal graphs via a finite forbidden‑subgraph list, and demonstrates that, under a modest independence‑number restriction, BQ‑freeness is also sufficient for a chordal graph to be a circular‑arc graph. The work deepens our understanding of the structural interplay between chordal and circular‑arc graphs and opens avenues for further research, such as extending the sufficiency result to larger independence numbers or exploring augmentation strategies that eliminate BQs while preserving chordality.