Graph traversal is all about exploring connections. Walks, paths, and cycles are ways to move through a graph, each with unique rules. These concepts help us understand how information or objects flow through networks.
Open and closed walks, along with length calculations, have real-world applications. From planning delivery routes to analyzing social networks, these ideas shape how we navigate interconnected systems in everyday life.
Fundamental Concepts of Graph Traversal
Walks, paths, and cycles
- Walk traverses graph sequence vertices and edges connecting consecutive vertices may repeat (Manhattan street grid)
- Path special walk no repeated vertices or edges except possibly first/last (GPS route)
- Cycle closed walk starts/ends same vertex no repeats except start/end minimum length 3 simple graphs (Monopoly board)
Open vs closed walks
- Open walk starts ends different vertices any length (train route with multiple stops)
- Closed walk forms loop starts ends same vertex minimum length 1 for self-loops (circular subway line)
Practical Applications
Identifying graph sequences
- Walk verification check consecutive vertex pairs connected by edge allow repetition (social network friend chain)
- Path verification valid walk no repeated vertices/edges except possibly first/last (delivery route)
- Cycle verification valid walk same start/end no repeats except start/end minimum length 3 simple graphs (yearly calendar)
Length of graph structures
- Length calculation count edges traversed weighted graphs sum edge weights (road network with distances)
- Walk length total edge traversals including repeats (tourist sightseeing route)
- Path length edges in path equals vertices minus one (flight itinerary)
- Cycle length edges/vertices in cycle minimum 3 simple graphs (Olympic rings)
- Relation to properties diameter longest shortest path between vertices girth shortest cycle (computer network topology)