Walks, Trails, and Paths

Graph Theory · Axiom Academy

LESSON Walks, Trails, and Paths Understanding fundamental movement concepts in graph theory 1. Walks: The Most General Movement A walk is a sequence of vertices where each consecutive pair of vertices is connected by an edge. Walks can repeat both vertices and edges. The animation below shows a walk that visits vertices multiple times and traverses some edges more than once. 2. Trails: Walks Without Repeated Edges A trail is a walk in which no edge is repeated. However, vertices may still be visited multiple times. Watch as the animation demonstrates a trail that visits some vertices multiple times but never uses the same edge twice. 3. Paths: Walks Without Repeated Vertices A path is a walk in which all vertices (and therefore all edges) are distinct. This is the most restrictive type of movement. The animation shows a path where each vertex is visited exactly once. The length of a walk, trail, or path is the number of edges it contains (not the number of vertices). The animation illustrates a closed walk and a cycle, highlighting the difference. Let's compare all three concepts side by side to solidify our understanding.

This is the written version of the interactive lesson above. See the full Graph Theory course.