  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.
