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).

Barrus, M. D.; Trenk, A. N.; and Whitman, R. "The Hereditary Closure of the Unigraphs." 23 Aug 2023. https://arxiv.org/abs/2308.12190Johnson,
R. H. "Simple Separable Graphs." Pacific J. Math.56,
143-158, 1975.Kleitman, D. J. and Li, S.-Y. "A Note on Unigraphic
Sequences." Stud. Appl. Math.54, 283-287, 1975.Koren,
M. "Pairs of Sequences with a Unique Realization by Bipartite Graphs."
J. Combin. Th. Ser. B21, 224-234, 1976.Koren, M. "Sequences
with a Unique Realization by Simple Graphs." J. Combin. Th. Ser. B21,
235-244, 1976.Li, S.-Y. R. "Graphic Sequences with Unique
Realizations." J. Combin. Th. Ser. B19, 42-68, 1975.Sloane,
N. J. A. Sequences A122423, A309757,
and A365548 in "The On-Line Encyclopedia
of Integer Sequences."Tyshkevich, R. "The Canonical Decomposition
of a Graph." Dokl. Akad. Nauk. BSSR24, 677-679, 1980.Tyshkevich,
R. "Decomposition of Graphical Sequences and Unigraphs." Disc. Math.220,
201-238, 2000.