Delta-matroid

In mathematics, a delta-matroid or Δ-matroid is a family of sets obeying an exchange axiom generalizing an axiom of matroids. A non-empty family of sets is a delta-matroid if, for every two sets $E$ and $F$ in the family, and for every element $e$ in their symmetric difference $E\mathrm{△}F$, there exists an $f\in E\mathrm{△}F$ such that $E\mathrm{△}\{e,f\}$ is in the family. For the basis sets of a matroid, the corresponding exchange axiom requires in addition that $e\in E$ and $f\in F$, ensuring that $E$ and $F$ have the same cardinality. For a delta-matroid, either of the two elements may belong to either of the two sets, and it is also allowed for the two elements to be equal.Template:R An alternative and equivalent definition is that a family of sets forms a delta-matroid when the convex hull of its indicator vectors (the analogue of a matroid polytope) has the property that every edge length is either one or the square root of two.

Delta-matroids were defined by André Bouchet in 1987.Template:R Algorithms for matroid intersection and the matroid parity problem can be extended to some cases of delta-matroids.Template:R

Delta-matroids have also been used to study constraint satisfaction problems.Template:R As a special case, an even delta-matroid is a delta-matroid in which either all sets have even number of elements, or all sets have an odd number of elements. If a constraint satisfaction problem has a Boolean variable on each edge of a planar graph, and if the variables of the edges incident to each vertex of the graph are constrained to belong to an even delta-matroid (possibly a different even delta-matroid for each vertex), then the problem can be solved in polynomial time. This result plays a key role in a characterization of the planar Boolean constraint satisfaction problems that can be solved in polynomial time.Template:R