Polypaths

Polypaths are a concept in theoretical computer science and graph theory, referring to a sequence of vertices in a graph such that each vertex in the sequence is adjacent to the next vertex. More specifically, a polypath can be understood as a walk in a graph that is not required to be simple; that is, it may revisit vertices and edges. The term "polypath" is not as commonly used as "walk," but it emphasizes the potential for repeated traversal of graph elements. A walk is formally defined as a sequence of vertices $v_0, v_1, \dots, v_k$ where for each $i$ from 0 to $k-1$, there is an edge between $v_i$ and $v_{i+1}$. The length of a polypath is the number of edges it contains, which is $k$. If the starting and ending vertices are the same, the polypath is called a closed polypath or a closed walk. If no vertices are repeated, it is a simple path. The study of polypaths is fundamental to understanding connectivity, reachability, and the structure of graphs. Algorithms that traverse graphs, such as breadth-first search and depth-first search, implicitly explore polypaths to discover graph properties. The existence and length of polypaths between any two vertices are key considerations in network analysis and algorithm design.