WhitePath Theorem
The whitepath theorem states that if there is a path between two vertices consisting of whitecolored edges, then there must exist a shortest path that is also white between those vertices.
This has applications in analyzing shortest paths with categorizations of edges.
For example, in a road network:
 White edges = highways
 Black edges = local roads
Then there exists a shortest highwayonly path between cities if any path exists using highways.
Java  Road network example:


C++ example  Graph coloring:


Python  Preferential attachment:


The whitepath theorem provides useful insights into structuring shortest paths in categorization problems.
The WhitePath Theorem is a concept in graph theory and is particularly useful in the study of depthfirst search (DFS) algorithms. According to this theorem, in a depthfirst forest produced by a DFS, there is a white path between vertex (u) and vertex (v) if and only if (u) is an ancestor of (v). A white path is a path where all vertices, except possibly the first, are white, meaning they are undiscovered. This theorem helps in analyzing and understanding properties related to depthfirst search in graphs.
The WhitePath Theorem is related to DepthFirst Search (DFS) in graph theory. The theorem states that in a DFS forest of a directed or undirected graph G, there is a path from vertex u to vertex v if and only if u is an ancestor of v in the DFS tree.
Visual Representation:
Imagine a directed graph as follows, where arrows show the direction from one vertex to another:
A > B < C
 
v v
D < E > F
Perform a DFS starting from vertex A. The DFS tree may look something like this:
A

B

D
In this DFS tree, according to the WhitePath Theorem, there should be a white path (i.e., a path discovered during the DFS) from A to B and from A to D. This matches our original graph, confirming the theorem.
Key Takeaway:
The WhitePath Theorem gives us a way to identify if there exists a path from one vertex to another by simply examining the DFS tree. This can be a powerful tool for understanding the structure of graphs and optimizing related algorithms.
Solution
Below are code examples that demonstrate the concept of the WhitePath Theorem using depthfirst search in Java, C++, and Python.
Java


C++


Python


Key Takeaways
 The WhitePath Theorem is directly related to the behavior of depthfirst search in graphs.
 It helps to identify whether there exists a white path between two vertices, aiding in graph analysis.
 The code examples use depthfirst search to illustrate the concept. A path between two vertices confirms the theorem.