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$\mathrm{with}\left(\mathrm{PolyhedralSets}\right)\:$

Graph of the tetrahedron's faces. The graph has one node in its top row (the tetrahedron), four nodes in its second row (the tetrahedron's faces), six nodes in its third row (the tetrahedron's edges), four nodes in its fourth row (the tetrahedron's vertices), and one node in its last row (the empty set).
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$t\u2254\mathrm{ExampleSets}:\mathrm{Tetrahedron}\left(\right)\:$$\mathrm{Graph}\left(t\right)$

The graph object can also be obtained using the output option, which can be further analyzed using the GraphTheory package.
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$c\u2254\mathrm{ExampleSets}:\mathrm{Cube}\left(\right)\:$$g\,p\u2254\mathrm{Graph}\left(c\,\'\mathrm{output}\'\=\left[\'\mathrm{graph}\'\,\'\mathrm{plot}\'\right]\right)\:$$p$

The ID's of the facets for a given node can be obtained from the graph.
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${\mathrm{GraphTheory}}_{\mathrm{Departures}}\left(g\,154\right)$

$\left[{109}{\,}{91}\right]$
 (1) 
The PolyhedralSets[Faces] command provides the means of access the faces based on their ID's from the graph.
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$f\u2254\mathrm{Faces}\left(c\,\'\mathrm{faceid}\'\=91\right)$

${f}{\u2254}{\{}\begin{array}{lll}{\mathrm{Coordinates}}& {\:}& \left[{{x}}_{{1}}{\,}{{x}}_{{2}}{\,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {\:}& \left[{{x}}_{{3}}{=}{\mathrm{1}}{\,}{{x}}_{{2}}{=}{\mathrm{1}}{\,}{{x}}_{{1}}{=}{\mathrm{1}}\right]\end{array}$
 (2) 