Two vertices are on a ______ if there is a sequences of vertices beginning with the first one and ending with the second one

Two vertices are on a path if there is a sequence of vertices beginning with the first one and ending with the second one. In graph theory, a path is a fundamental concept that describes a way of moving through a graph by traversing the edges that connect the vertices. A path can be defined as an ordered sequence of vertices where each adjacent pair is connected by an edge in the graph.

The concept of paths can vary depending on whether the graph is directed or undirected. In directed graphs, paths must follow the direction of the edges, while in undirected graphs, paths can traverse edges in any direction. The length of a path is often defined as the number of edges it contains. Paths are crucial in various applications such as network design, data structure traversal, and algorithm development, as they represent relationships and connections between different entities represented by the vertices.

In summary, when discussing paths between vertices in a graph, it denotes the connective journey one can make through a series of vertices linked by edges, reflecting the structure and dynamics of the graph itself.