reduction = truefalse

There are two (potentially) different graphs that this command can generate: the graph where the arrows form the transitive reduction of the ordering of P, and the graph where the arrows form the transitive closure of the ordering. By default, ToGraph returns the transitive reduction graph (the Hasse diagram). When passed the option reduction = false, it returns the transitive closure graph. Passing the option reduction or reduction = true explicitly selects the default transitive reduction graph.

setcoordinates = truefalse

When constructing the transitive reduction graph, Maple will by default compute a suitable layout of the vertices, so that a subsequent call to GraphTheory:-DrawGraph can use that layout. To disable this feature, you can pass the option setcoordinates = false. Passing the option setcoordinates or setcoordinates = true explicitly selects the default of computing the coordinates, but it cannot be passed if computing the transitive closure graph.

graded = truefalse

prefer = low or high

If Maple computes a layout for the graph, the options graded and prefer are used by the PartiallyOrderedSets:-DrawGraph command to fine-tune the layout. They are explained on that command's help page. The options cannot be used if computing the transitive closure graph.

dgopts (not a literal option name)

If passing any extra options not mentioned before (represented by dgopts in the calling sequence above), they are passed on to the GraphTheory:-DrawGraph command.

The command ToGraph(P) returns the graph representation of the partially ordered set P as a graph object of the GraphTheory package. Vertices of the graph are elements of P, and directed edges represent relations between the elements.

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

$\mathrm{leq}≔\mathrm{`<=`}\&colon;$

$S≔\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\right\}\&colon;$$\mathrm{poset1}≔\mathrm{PartiallyOrderedSet}\left(S\&comma;\mathrm{leq}\right)$

$\mathrm{poset1}≔\mathrm{<\; a\; poset\; with\; 5\; elements\; >}$

$G≔\mathrm{ToGraph}\left(\mathrm{poset1}\right)$

$G≔\mathrm{Graph\; 1:\; a\; directed\; graph\; with\; 5\; vertices\; and\; 4\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(G\right)$

$\mathrm{lneq}≔\mathrm{`<`}\&colon;$$\mathrm{poset1\_1}≔\mathrm{PartiallyOrderedSet}\left(S\&comma;\mathrm{lneq}\right)$

$\mathrm{poset1\_1}≔\mathrm{<\; a\; poset\; with\; 5\; elements\; >}$

$\mathrm{G1\_1}≔\mathrm{ToGraph}\left(\mathrm{poset1\_1}\right)$

$\mathrm{G1\_1}≔\mathrm{Graph\; 2:\; a\; directed\; graph\; with\; 5\; vertices\; and\; 4\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G1\_1}\right)$

$\mathrm{divisibility}≔\left(x\&comma;y\right)\mapsto \mathrm{irem}\left(y\&comma;x\right)=0\&colon;$$T≔\left\{3\&comma;4\&comma;5\&comma;6\&comma;7\&comma;8\&comma;9\right\}\&colon;$

$\mathrm{poset2}≔\mathrm{PartiallyOrderedSet}\left(T\&comma;\mathrm{divisibility}\right)$

$\mathrm{poset2}≔\mathrm{<\; a\; poset\; with\; 7\; elements\; >}$

$\mathrm{G2}≔\mathrm{ToGraph}\left(\mathrm{poset2}\right)$

$\mathrm{G2}≔\mathrm{Graph\; 3:\; a\; directed\; graph\; with\; 7\; vertices\; and\; 3\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G2}\right)$

$\mathrm{divisibNE}≔\left(x\&comma;y\right)\mapsto \mathrm{irem}\left(y\&comma;x\right)=0\mathbf{and}y\ne x\&colon;$

$\mathrm{poset2\_1}≔\mathrm{PartiallyOrderedSet}\left(T\&comma;\mathrm{divisibNE}\&comma;\mathrm{reflexive}=\mathrm{checkfalse}\right)$

$\mathrm{poset2\_1}≔\mathrm{<\; a\; poset\; with\; 7\; elements\; >}$

$\mathrm{G2\_1}≔\mathrm{ToGraph}\left(\mathrm{poset2\_1}\right)$

$\mathrm{G2\_1}≔\mathrm{Graph\; 4:\; a\; directed\; graph\; with\; 7\; vertices\; and\; 3\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G2\_1}\right)$

$U≔\left\{1\&comma;2\&comma;3\right\}\&colon;$

$\mathrm{poset3}≔\mathrm{PartiallyOrderedSet}\left(U\&comma;\mathrm{leq}\&comma;\mathrm{reflexive}=\mathrm{checktrue}\right)$

$\mathrm{poset3}≔\mathrm{<\; a\; poset\; with\; 3\; elements\; >}$

$\mathrm{G3}≔\mathrm{ToGraph}\left(\mathrm{poset3}\right)$

$\mathrm{G3}≔\mathrm{Graph\; 5:\; a\; directed\; graph\; with\; 3\; vertices\; and\; 2\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G3}\right)$

$\mathrm{poset3\_1}≔\mathrm{PartiallyOrderedSet}\left(U\&comma;\mathrm{lneq}\&comma;\mathrm{reflexive}=\mathrm{useclosure}\right)$

$\mathrm{poset3\_1}≔\mathrm{<\; a\; poset\; with\; 3\; elements\; >}$

$\mathrm{G3\_1}≔\mathrm{ToGraph}\left(\mathrm{poset3\_1}\right)$

$\mathrm{G3\_1}≔\mathrm{Graph\; 6:\; a\; directed\; graph\; with\; 3\; vertices\; and\; 2\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G3\_1}\right)$

$X≔\left\{4\&comma;5\&comma;6\right\}\&colon;$$\mathrm{poset3\_2}≔\mathrm{PartiallyOrderedSet}\left(X\&comma;\mathrm{leq}\&comma;\mathrm{reflexive}=\mathrm{checktrue}\right)$

$\mathrm{poset3\_2}≔\mathrm{<\; a\; poset\; with\; 3\; elements\; >}$

$\mathrm{G3\_2}≔\mathrm{ToGraph}\left(\mathrm{poset3\_2}\right)$

$\mathrm{G3\_2}≔\mathrm{Graph\; 7:\; a\; directed\; graph\; with\; 3\; vertices\; and\; 2\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G3\_2}\right)$

$\mathrm{adjMatrix4}≔\mathrm{Matrix}\left(\left[\left[1\&comma;1\&comma;1\&comma;1\&comma;1\right]\&comma;\left[0\&comma;1\&comma;1\&comma;1\&comma;1\right]\&comma;\left[0\&comma;0\&comma;1\&comma;1\&comma;1\right]\&comma;\left[0\&comma;0\&comma;0\&comma;1\&comma;1\right]\&comma;\left[0\&comma;0\&comma;0\&comma;0\&comma;1\right]\right]\right)$

$\mathrm{adjMatrix4}≔\left[\begin{array}{ccccc}1& 1& 1& 1& 1\\ 0& 1& 1& 1& 1\\ 0& 0& 1& 1& 1\\ 0& 0& 0& 1& 1\\ 0& 0& 0& 0& 1\end{array}\right]$

$\mathrm{poset4}≔\mathrm{PartiallyOrderedSet}\left(\mathrm{convert}\left(S\&comma;'\mathrm{list}'\right)\&comma;\mathrm{adjMatrix4}\right)$

$\mathrm{poset4}≔\mathrm{<\; a\; poset\; with\; 5\; elements\; >}$

$\mathrm{G4}≔\mathrm{ToGraph}\left(\mathrm{poset4}\right)$

$\mathrm{G4}≔\mathrm{Graph\; 8:\; a\; directed\; graph\; with\; 5\; vertices\; and\; 4\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G4}\right)$

$\mathrm{adjList5}≔\mathrm{map2}\left(\mathrm{map}\&comma;\mathrm{`+`}\&comma;\mathrm{Array}\left(\left[\left\{1\&comma;4\&comma;7\right\}\&comma;\left\{2\&comma;6\right\}\&comma;\left\{3\right\}\&comma;\left\{4\right\}\&comma;\left\{5\right\}\&comma;\left\{6\right\}\&comma;\left\{7\right\}\right]\right)\&comma;2\right)$

$\mathrm{adjList5}≔\left[\begin{array}{ccccccc}\left\{3\&comma;6\&comma;9\right\}& \left\{4\&comma;8\right\}& \left\{5\right\}& \left\{6\right\}& \left\{7\right\}& \left\{8\right\}& \left\{9\right\}\end{array}\right]$

$\mathrm{poset5}≔\mathrm{PartiallyOrderedSet}\left(\mathrm{convert}\left(T\&comma;'\mathrm{list}'\right)\&comma;\mathrm{adjList5}\right)$

$\mathrm{poset5}≔\mathrm{<\; a\; poset\; with\; 7\; elements\; >}$

$\mathrm{G5}≔\mathrm{ToGraph}\left(\mathrm{poset5}\right)$

$\mathrm{G5}≔\mathrm{Graph\; 9:\; a\; directed\; graph\; with\; 7\; vertices\; and\; 3\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G5}\right)$

$G≔\mathrm{GraphTheory}:-\mathrm{Graph}\left(\mathrm{directed}\&comma;\left[1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\right]\&comma;\left\{\left[1\&comma;1\right]\&comma;\left[1\&comma;2\right]\&comma;\left[1\&comma;3\right]\&comma;\left[1\&comma;4\right]\&comma;\left[1\&comma;5\right]\&comma;\left[1\&comma;6\right]\&comma;\left[2\&comma;2\right]\&comma;\left[2\&comma;4\right]\&comma;\left[2\&comma;6\right]\&comma;\left[3\&comma;3\right]\&comma;\left[3\&comma;5\right]\&comma;\left[3\&comma;6\right]\&comma;\left[4\&comma;4\right]\&comma;\left[4\&comma;6\right]\&comma;\left[5\&comma;5\right]\&comma;\left[5\&comma;6\right]\&comma;\left[6\&comma;6\right]\right\}\right)$

$G≔\mathrm{Graph\; 10:\; a\; directed\; graph\; with\; 6\; vertices,\; 11\; arcs,\; and\; 6\; self-loops}$

$\mathrm{poset6}≔\mathrm{PartiallyOrderedSet}\left(G\right)$

$\mathrm{poset6}≔\mathrm{<\; a\; poset\; with\; 6\; elements\; >}$

$\mathrm{G6}≔\mathrm{ToGraph}\left(\mathrm{poset6}\right)$

$\mathrm{G6}≔\mathrm{Graph\; 11:\; a\; directed\; graph\; with\; 6\; vertices\; and\; 6\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G6}\right)$

$\mathrm{poset7}≔\mathrm{PartiallyOrderedSet}\left(\mathrm{convert}\left(U\&comma;'\mathrm{list}'\right)\&comma;⟨⟨1\left|1\right|0⟩\&comma;⟨0\left|1\right|1⟩\&comma;⟨0\left|0\right|1⟩⟩\&comma;\mathrm{input}=\mathrm{transitivereduction}\right)$

$\mathrm{poset7}≔\mathrm{<\; a\; poset\; with\; 3\; elements\; >}$

$\mathrm{G7}≔\mathrm{ToGraph}\left(\mathrm{poset7}\right)$

$\mathrm{G7}≔\mathrm{Graph\; 12:\; a\; directed\; graph\; with\; 3\; vertices\; and\; 2\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G7}\right)$

$\mathrm{poset8}≔\mathrm{PartiallyOrderedSet}\left(\left[1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\right]\&comma;\mathrm{Array}\left(\left[\left\{1\&comma;2\&comma;3\right\}\&comma;\left\{2\&comma;4\right\}\&comma;\left\{3\&comma;5\right\}\&comma;\left\{4\&comma;6\right\}\&comma;\left\{5\&comma;6\right\}\&comma;\left\{6\right\}\right]\right)\&comma;\mathrm{input}=\mathrm{transitivereduction}\right)$

$\mathrm{poset8}≔\mathrm{<\; a\; poset\; with\; 6\; elements\; >}$

$\mathrm{G8}≔\mathrm{ToGraph}\left(\mathrm{poset8}\right)$

$\mathrm{G8}≔\mathrm{Graph\; 13:\; a\; directed\; graph\; with\; 6\; vertices\; and\; 6\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G8}\right)$

$t≔\mathrm{PolyhedralSets}:-\mathrm{ExampleSets}:-\mathrm{Octahedron}\left(\right)$

$t≔\&lcub;\begin{array}{lll}\mathrm{Coordinates}& \&colon;& \left[x_1\&comma;x_2\&comma;x_3\right]\\ \mathrm{Relations}& \&colon;& \left[-x_1-x_2-x_3\le 1\&comma;-x_1-x_2+x_3\le 1\&comma;-x_1+x_2-x_3\le 1\&comma;-x_1+x_2+x_3\le 1\&comma;x_1-x_2-x_3\le 1\&comma;x_1-x_2+x_3\le 1\&comma;x_1+x_2-x_3\le 1\&comma;x_1+x_2+x_3\le 1\right]\end{array}$

$d≔\mathrm{PolyhedralSets}:-\mathrm{Dimension}\left(t\right)$

$d≔3$

$\mathrm{t\_faces}≔\left\{\mathrm{seq}\left(\mathrm{op}\left(\mathrm{PolyhedralSets}:-\mathrm{Faces}\left(t\&comma;\mathrm{dimension}=i\right)\right)\&comma;i=-0..d\right)\right\}\&colon;$

$\mathrm{t\_faces}≔\mathrm{t\_faces}\mathbf{union}\left\{\mathrm{PolyhedralSets}:-\mathrm{ExampleSets}:-\mathrm{EmptySet}\left(d\right)\right\}\&colon;$

$\mathrm{FL}≔\mathrm{convert}\left(\mathrm{t\_faces}\&comma;\mathrm{list}\right)\&colon;$

inclusion := proc(x,y) PolyhedralSets:-`subset`(FL[x],FL[y]) end proc:

$\mathrm{polyhedral\_poset}≔\mathrm{PartiallyOrderedSet}\left(\left\{\mathrm{seq}\left(i\&comma;i=1..\mathrm{nops}\left(\mathrm{FL}\right)\right)\right\}\&comma;\mathrm{inclusion}\right)$

$\mathrm{polyhedral\_poset}≔\mathrm{<\; a\; poset\; with\; 28\; elements\; >}$

$\mathrm{G\_POLY}≔\mathrm{ToGraph}\left(\mathrm{polyhedral\_poset}\right)$

$\mathrm{G\_POLY}≔\mathrm{Graph\; 14:\; a\; directed\; graph\; with\; 28\; vertices\; and\; 62\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G\_POLY}\right)$

$M≔\mathrm{Matrix}\left(\left[\left[1\&comma;1\&comma;1\&comma;1\&comma;1\right]\&comma;\left[0\&comma;1\&comma;1\&comma;0\&comma;1\right]\&comma;\left[0\&comma;0\&comma;1\&comma;0\&comma;1\right]\&comma;\left[0\&comma;0\&comma;0\&comma;1\&comma;1\right]\&comma;\left[0\&comma;0\&comma;0\&comma;0\&comma;1\right]\right]\right)\&colon;$

$\mathrm{poset9}≔\mathrm{PartiallyOrderedSet}\left(\left[\mathrm{seq}\left(1..5\right)\right]\&comma;M\right)$

$\mathrm{poset9}≔\mathrm{<\; a\; poset\; with\; 5\; elements\; >}$

$\mathrm{G9}≔\mathrm{ToGraph}\left(\mathrm{poset9}\right)$

$\mathrm{G9}≔\mathrm{Graph\; 15:\; a\; directed\; graph\; with\; 5\; vertices\; and\; 5\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G9}\right)$

$Z≔\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;10\&comma;12\&comma;15\&comma;20\&comma;30\&comma;60\right\}$

$Z≔\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\&comma;6\&comma;10\&comma;12\&comma;15\&comma;20\&comma;30\&comma;60\right\}$

$\mathrm{poset10}≔\mathrm{PartiallyOrderedSet}\left(Z\&comma;\mathrm{divisibility}\right)$

$\mathrm{poset10}≔\mathrm{<\; a\; poset\; with\; 12\; elements\; >}$

$\mathrm{G10\_C}≔\mathrm{ToGraph}\left(\mathrm{poset10}\&comma;\mathrm{reduction}=\mathrm{false}\right)$

$\mathrm{G10\_C}≔\mathrm{Graph\; 16:\; a\; directed\; graph\; with\; 12\; vertices\; and\; 42\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G10\_C}\right)$

$\mathrm{G10\_RNC}≔\mathrm{ToGraph}\left(\mathrm{poset10}\&comma;\mathrm{setcoordinates}=\mathrm{false}\right)$

$\mathrm{G10\_RNC}≔\mathrm{Graph\; 17:\; a\; directed\; graph\; with\; 12\; vertices\; and\; 20\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G10\_RNC}\right)$

$\mathrm{G10\_RC}≔\mathrm{ToGraph}\left(\mathrm{poset10}\right)$

$\mathrm{G10\_RC}≔\mathrm{Graph\; 17:\; a\; directed\; graph\; with\; 12\; vertices\; and\; 20\; arcs}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{G10\_RC}\right)$

Richard P. Stanley: Enumerative Combinatorics 1. 1997, Cambridge Studies in Advanced Mathematics. Vol. 49. Cambridge University Press.

The PartiallyOrderedSets[ToGraph] command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.