Category:Element of Matroid Base and Circuit has Substitute

This category contains pages concerning Element of Matroid Base and Circuit has Substitute:

Let $M = \struct {S, \mathscr I}$ be a matroid.

Let $B \subseteq S$ be a base of $M$.

Let $C \subseteq S$ be a circuit of $M$.

Let $x \in B \cap C$.

Then:

$\exists y \in C \setminus B : \paren{B \setminus \set x} \cup \set y$ is a base of $M$

That is, there exists $y \in C \setminus B$ such that substituting $y$ for $x$ in $B$ is another base.