Graphs

$G=(V,E)$ where $V \neq \empty$

$V = \{x_1,…x_6\}$

$E = \{(x_1, x_2), …\}$

$|V|$ and $|E|$ denote their cardinalities

Definition: A matching is a subgraph of G where every $v\in V$ has degree 1.

graph TD;
	1 --- 2
	3 --- 5
	4 --- 5
	1 --- 8
	1 --- 7
	1 --- 6
	6 --- 5

In this graph a matching only exists for $(\{1,6\},\{2,5\})$.

A matching is perfect if it has size $\dfrac{|V|}{2}$.

Assign positive weights to edges (lower is better):

flowchart LR
    b1 === g1
		b2 === g3
		b3 === g2
		b4 === g4
		b1 --- g2
		b2 --- g4
		b3 --- g1
		b4 --- g1

This graph has a perfect matching (in bold).

Definition: The weight of a perfect matching its sum of the weights of the paired edges.

Definition: A min weight perfect matching has the minimum weight among all possible perfect matchings.

graph LR
	David ---|10| Angelina
	David ---|16| Jennifer
	Brad ---|5| Angelina
	Brad ---|10| Jennifer

In this example, a perfect matching with minimum weight uses the edges with weights 10.