Dual - Maple Help

Matroids

 Dual
 construct the dual of a matroid

 Calling Sequence Dual(M)

Parameters

 M -

Description

 • The dual ${M}^{*}$ of a matroid $M$ is itself a matroid on the same ground set. The bases of ${M}^{*}$ are the complements of bases of $M$.
 • The operations of deletion and contraction are dual to each other. That is, ${\left(M\setminus \left\{e\right\}\right)}^{*}$ is isomorphic to ${M}^{*}/\left\{e\right\}$.
 • The Tutte polynomial of the dual of a matroid simply swaps the roles of the variables.
 • The dual of a representable matroid is the matroid on its orthogonal complement. More precisely, if $M$ is the matroid representable by the columns of a matrix $A$, and $B$ is a matrix whose rows form the orthogonal complement to the rows of $A$, then ${M}^{*}$ is the matroid representable by the columns of $B$.
 • Given a planar graph $G$, the dual of the matroid underlying $G$ is the graph dual of $G$.
 • The dual of the dual of a matroid $M$ is itself.

Examples

 > $\mathrm{with}\left(\mathrm{Matroids}\right):$

Consider the following matroid, $M$.

 > $A≔\mathrm{Matrix}\left(\left[\left[1,1,1,0,0,0\right],\left[0,0,0,1,1,1\right],\left[1,0,0,1,0,0\right],\left[0,1,0,0,1,0\right],\left[0,0,1,0,0,2\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{cccccc}{1}& {1}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {1}& {1}\\ {1}& {0}& {0}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}& {1}& {0}\\ {0}& {0}& {1}& {0}& {0}& {2}\end{array}\right]$ (1)
 > $M≔\mathrm{Matroid}\left(A\right)$
 ${M}{≔}⟨\begin{array}{c}{thⅇ linⅇar matroiⅆ whosⅇ grounⅆ sⅇt is thⅇ sⅇt of column vⅇctors of thⅇ matrix:}\\ \left[\begin{array}{cccccc}{1}& {1}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {1}& {1}\\ {1}& {0}& {0}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}& {1}& {0}\\ {0}& {0}& {1}& {0}& {0}& {2}\end{array}\right]\end{array}⟩$ (2)
 > $B≔\mathrm{Bases}\left(M\right)$
 ${B}{≔}\left[\left\{{1}{,}{2}{,}{3}{,}{4}{,}{6}\right\}{,}\left\{{1}{,}{2}{,}{3}{,}{5}{,}{6}\right\}{,}\left\{{1}{,}{3}{,}{4}{,}{5}{,}{6}\right\}{,}\left\{{2}{,}{3}{,}{4}{,}{5}{,}{6}\right\}\right]$ (3)

The bases of the dual of $M$ are the complements of the bases of $M$.

 > $\mathrm{M2}≔\mathrm{Dual}\left(M\right)$
 ${\mathrm{M2}}{≔}⟨{a matroiⅆ on 6 ⅇlⅇmⅇnts with 4 basⅇs of sizⅇ 1}⟩$ (4)
 > $\mathrm{B2}≔\mathrm{Bases}\left(\mathrm{M2}\right)$
 ${\mathrm{B2}}{≔}\left[\left\{{1}\right\}{,}\left\{{2}\right\}{,}\left\{{4}\right\}{,}\left\{{5}\right\}\right]$ (5)

The Tutte polynomial of $\mathrm{M2}$ is the Tutte polynomial of $M$ with the variables swapped.

 > $T≔\mathrm{TuttePolynomial}\left(M,x,y\right)$
 ${T}{≔}{{x}}^{{5}}{+}{{x}}^{{4}}{+}{{x}}^{{3}}{+}{{x}}^{{2}}{}{y}$ (6)
 > $\mathrm{T2}≔\mathrm{TuttePolynomial}\left(\mathrm{M2},y,x\right)$
 ${\mathrm{T2}}{≔}{{x}}^{{5}}{+}{{x}}^{{4}}{+}{{x}}^{{3}}{+}{{x}}^{{2}}{}{y}$ (7)

References

 James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). New York: Oxford University Press. 2006.