A unigraphic graph (or simply a "unigraph") is a graph that is isomorphic to every graph having that degree sequence. All graphs on four are fewer vertices are unigraphic.
The numbers of unigraphs on 
, 2, ... nodes are 1, 2, 4, 11, 28, 72, 170, 407, 956, 2252
... (OEIS A122423), and the numbers of these
that are connected are 1, 1, 2, 6, 16, 42, 96, 234, 546, 1292, ... (OEIS A365548).
The numbers of connected graphs that have distinct degree sequences among all connected
graphs with 
nodes for 
,
2, ... are 1, 1, 1, 2, 6, 17, 45, 99, 238, 549, 1296, ... (OEIS A309757).
There are therefore 6 graphs on 5 vertices which are not unigraphic, including the path graph 
and 
which share the degree
sequence 
.
If a graph 
is a unigraph, then its graph complement 
is also a unigraph (Barrus et al.
2023).
An algorithm for recognizing unigraphicity in linear time is described by Kleitman (1975). While the class of unigraphs has no forbidden induced subgraph characterization,
it can be characterized via decomposition (Barrus et al. 2023). In particular,
a graph 
is a unigraph iff each component of its Tyshkevich decomposition
is a unigraph (Tyshkevich 2000, Barrus et al. 2023).
The only regular unigraphs are the complete graphs 
,
empty graphs 
, ladder rung graphs 
, cocktail
party graphs 
,
and the cycle graph 
(Johnson 1975, Koren 1976, Tyshkevich 2000).
