A triangle-replaced graph 
is a cubic graph in which
each vertex is replaced by a triangle graph such
that each vertex of the triangle is connected to one of the originally adjacent vertices
of a graph 
.
The triangle-replaced Coxeter graph appears as an exceptional graph in conjectures about nonhamiltonian vertex-transitive graphs, H-*-connected graphs, and Hamilton decompositions.
Bryant and Dean (2014) consider a generalization to a 
-replaced graph, in which the vertices of a 
-regular graph are replaced by copies of the complete
graph 
.
Such graphs provide counterexamples to the conjecture that there are a finite number
of connected vertex-transitive
graphs that have no Hamilton decomposition.
The smallest counterexample is given by the 
-replaced graph obtained from the multigraph obtained from
the cubical graph 
by doubling its edges.
Special cases of triangle-replaced graphs are summarized in the following table.
