A chordal ring is an augmented ring, or a *circulant graph* with
a chord of length 1. Formally it is defined by the pair (*n*,*L*) where
*n* is the number of nodes of the ring, and *L* is the set of chords,
.
Each chord
connects
every pair of nodes of the ring that are at distance *l* in the ring.
We denote by
,
,
the
chordal ring defined by
.
The dimension of
is *t*+1. Let us emphasize that the degree of
chordal rings is 2*t* in general, except whenever there is a chord of
length *n*/2. In this case *n* is even and the degree is 2*t*-1.

A ``natural'' labeling of the nodes for chordal rings is the cyclic labeling: Consecutive labeling of nodes around the ring. This labeling is not always the best possible one. For instance, [KL95] showed that has no shortest path 1-IRS with cyclic labeling whereas it is of compactness 1. In fact it is possible to characterize graphs supporting a shortest path 1-IRS according a cyclic labeling of nodes.

By the use of the Ádám property, [Man98] showed that is isomorphic to if and only if . For instance , which has no shortest path 1-IRS with cyclic labeling (cf. [KL95]), is isomorphic to which satisfies Theorem 26. Therefore, .

Let
.
Since
is of diameter *i*
(cf. [BCH95]), it follows that 1-
,
for
every .
(Actually,
is the chordal ring of degree 4
with the largest number of nodes for a given diameter). Moreover,
in [NO97], it is shown that 1-
.
Also, the
smallest chordal ring known to be of compactness greater than 1 is
,
and 1-
.
[NO97] showed that
there exists some chordal rings, ,
where every shortest path
IRS using a cyclic labeling requires
intervals, and also
that
has no shortest path 1-IRS of stretch factor
with a cyclic labeling. Other results about directed chordal rings
are mentioned in [KL95].

- Is there some chordal ring of compactness ?

- Characterize chordal rings of compactness

For two chords we have:

And for higher dimensions,

The latter result can be easily generalized to
when
,
and
for every *i*.

In [FGS94] is presented several results concerning IRS without the disjointness property of the labels assigned to the arcs, i.e., with Condition 2b of Definition 1 relaxed. See Paragraph 4.3 for a discussion of these results.

Note also that most of the results extend to circulant graphs in general, i.e., does not need to specify the chord of length 1.

- Is there a general upper bound for the compactness of chordal rings of dimensions