  Using a link between graph theory and the geometry hosting higher order
topological matter, we fill part of the missing results in the engineering of
domain walls supporting gapless states for systems with three vertical hinges.
The skeleton matrices which house the particle states responsible for the
physical properties are classified by the Euler characteristic into three sets
with topological index $\chi =0,1,2.$ A tri-hinge hamiltonian model invariant
under the composite $\boldsymbol{M}_{1}\boldsymbol{T}$,
$\boldsymbol{M}_{2}\boldsymbol{T}$, $\boldsymbol{M}_{3}\boldsymbol{T}$ is
built. In this framework, $\boldsymbol{T}$ is the time reversing symmetry
obeying $\boldsymbol{T}^{2}=-I$ and the $\boldsymbol{M}_{i}$'s are the
generators of the three reflections of the dihedral $\mathbb{D}_{3}$ symmetry
of triangle. To capture the tri-hinge states, candidate materials are
suggested, thus opening up a variety of possibilities for investigating and
designing robust materials against disorder and deformation.
