A chordless graph is a simple graph possessing no chords.
A chordal graph (which possesses no chordless cycles) is not the same as (or converse of) a chordless graph (which possesses no chords).
For example, the square graph 
is chordless but not chordal,
the diamond graph and tetrahedral
graph 
are chordal but not chordless, and empty
graphs 
,
path graphs 
, and the triangle graph 
are both chordal
and chordless.
A biconnected graph is chordless iff it is minimally 2-connected, meaning that deletion of any edge makes it no longer biconnected (Dirac 1967, Plummer 1968).
Aboulker et al. (2012) showed that every chordless graph on at least two vertices has at least two vertices of degree at most 2. Since every subgraph of a chordless graph is chordless, the graph degeneracy of a chordless graph is therefore at most 2. Equivalently, the 3-core of every chordless graph is empty. The converse is false: the diamond graph has degeneracy 2 and hence an empty 3-core, but is not chordless.
The numbers of connected simple chordless graphs on 
, 2, ... nodes are 1, 1, 2, 4, 10, 27, ... (OEIS A287693),
the first few of which are illustrated above.
The numbers of not-necessarily connected simple chordless graphs on 
, 2, ... nodes are 1, 2, 4, 9, 21, 56, ... (OEIS A287694),
the first few of which are illustrated above.
