  The problem of graph reconstruction has been studied in its various forms
over the years. In particular, the Reconstruction Conjecture, proposed by Ulam
and Kelly in 1942, has attracted much research attention and yet remains one of
the foremost unsolved problems in graph theory. Recently, Bastide, Cook,
Erickson, Groenland, Kreveld, Mannens, and Vermeulen proposed a new model of
partial information, where we are given the set of connected triples T_3, which
is the set of 3-subsets of the vertex set that induce connected subgraphs. They
proved that reconstruction is unique within the class of triangle-free graphs,
2-connected outerplanar graphs, and maximal planar graphs. They also showed
that almost every graph can be uniquely reconstructed from their connected
triples. However, little is known about other classes of non-triangle-free
graphs within which reconstruction can occur uniquely, nor do we understand
what kind of graphs can be uniquely reconstructed from their connected triples
without assuming anything about the classes of graphs to which they belong.
  The main result of this paper is a complete characterization of all graphs
that can be uniquely reconstructed from their connected triples T_3. We also
show that reconstruction from T_3 is unique within the class of regular planar
graphs, 5-connected planar graphs, certain strongly regular graphs, and
complete multi-partite graphs, whereas it is not unique for the class of
k-connected planar graphs with k less or equal to 4, Eulerian graphs, or
Hamiltonian graphs.
