The circuit of figure A has 4 essential nodes. In order for this circuit to be analyzed, (4-1=3), 3 equations are needed. 

figure A : example circuit

Figure B demonstrates  the branch currents of figure A.

figure B

On figure C, the 4 nodes of the circuit are numbered, and a tree  joins all the nodes together creating a path.


figure C

A tree is a set of connecting branches that connects every node to every other node, passing only once from each node (does not form a loop)

Some example trees of figure A can be:


              i3-i4-i6                                  i2-i4-i5                                


Once a tree is defined, the remaining branches are called links or chords. A collection of these links is called a complementary tree or cotree.

A graph is the redrawing of the circuit with each branch represented by a line, like figure B & figure C.


figure B                                                figure C

A cut-set is the smallest set of branches which are removed  from the graph in order to divide the graph into two groups of nodes. Some example cut-sets, are shown in figure D(i) & D(ii) below:


   cut-set: i1-i2-i3-i5-i6                      cut-set: i1-i2-i4-i6 
  figure D(i)                                        figure D(ii)

A fundamental loop  is a loop which contains one and only one link. By observing figure E:


                                               figure E      

fundamental loops: i2-i4-i3, i4-i6-i5, i1-i4-i3,


A fundamental cut-set  is a cut-set which contains one and only one tree branch. By observing figure E above, an example of a fundamental cut-set is shown on figure F & G


            figure F                                        

fundamental cut-set: i1-i2-i3-i7   

                                     figure G

fundamental cut-set: i1-i2-i4-i5-i7