To prove the breath first search properties in an undirected graph.

In the undirected graph, breath first search follow properties as:

Suppose in the undirected graph an edge ( u , v ) is consider as forward or back edge then vertex ‘ u’ should occur before vertex ‘ v’ and edges associated with ‘ u’ should be explore before any other edge in breath first search. Thus, the edge ( u , v ) can be considered as the tree edge because there is no forward and backward edge in undirected graph BFS traversal.

In the breath first search, an edge must be a tree if [v.π]=u and this can be possible if [v.d]=[u.d]+1 . In this values of [ u.d ] and [ v.d ] cannot be changed after initialization. So, in the breath first search [v.d]=[u.d]+1 .

Therefore for every tree edge ( u, v ) in breath first search, [v.d]=[u.d]+1 .

If vertex ‘ u’ visit before vertex ‘ v’ , and edge ( u, v ) is cross edge. As the properties of tree edge and cross edge, vertex ‘ u’ visit before vertex ‘ v’ that means vertex ‘ v’ must be in the queue. Hence, from the given statement, it will follow the condition for the cross edge ( u, v ):

Therefore, [v.d]=[u.d]+1 or [v.d]=[u.d].

To prove the breath first search properties in the directed graph.

Breath first search properties in the directed graph as follows:

An edge ( u, v ) is not the tree edge if the given graph G is directed. So, the graph will not contain the forward edge ( u, v ).

In the breath first search, an edge must be a tree if [v.π]=u and this can be possible if [v.d]=[u.d]+1 . In this values of [ u.d ] and [ v.d ] cannot be change after initialization. So, in the breath first search [v.d]=[u.d]+1 . Therefore, for every tree edge ( u, v ) in breath first search, [v.d]=[u.d]+1 .

If vertex ‘ u’ visit before vertex ‘ v’ , and edge ( u, v ) is cross edge. As the properties of tree edge and cross edge, vertex ‘ u’ visit before vertex ‘ v’ that means vertex ‘ v’ must be in the queue. Hence, from the given statement, it will follow the condition for the cross edge ( u, v ):

Therefore, [v.d]=[u.d]+1 or [v.d]=[u.d] .

By seeing the above statements, it is clear for all vertices, 0≤[v.d] .

In the breath first search, back edge (u, v) where vertex ‘v’ is the ancestor of vertex ‘u’ and from the given vertices if the drawn edge is larger than the other. Hence, [v.d]≤[u.d] .

Therefore, for every back edge (u, v), 0≤[v.d]≤[u.d] .