Matroid Unique Circuit Property/Corollary
Jump to navigation
Jump to search
Theorem
Let $M = \struct {S, \mathscr I}$ be a matroid.
Let $B$ be a base of $M$.
Let $x \in S \setminus B$.
Then there exists a unique circuit $C$ such that:
- $x \in C \subseteq B \cup \set x$
That is, $C$ is the fundamental circuit of $x$ in $B$.
Proof
From Union of Matroid Base with Element of Complement is Dependent:
- $B \cup \set x$ is dependent.
From Matroid Unique Circuit Property there exists a unique circuit $C$ such that:
- $x \in C \subseteq B \cup \set x$
$\blacksquare$
Sources
- 1976: Dominic Welsh: Matroid Theory ... (previous) ... (next) Chapter $1.$ $\S 9.$ Circuits