Cycles

Graph Theory · Axiom Academy

Understanding closed paths and their fundamental role in graph structure The path starts and ends at the same vertex At least 3 vertices are involved (for simple graphs) Watch as we construct cycles of different lengths. Notice how the path returns to its starting point, forming a closed loop. We use the notation Cₙ to represent a cycle with n vertices (and therefore n edges). The girth of a graph is the length of its shortest cycle. The animation below shows how to identify the shortest cycle in a graph. Finding girth is important for understanding graph structure and connectivity. To find a cycle, we can trace a path through the graph. When we encounter a vertex we've already visited, we've found a cycle! The animation demonstrates a depth-first search approach to cycle detection. We mark vertices as visited (colored) and watch for back edges that create cycles. A graph with no cycles is called acyclic . Connected acyclic graphs are called trees , and disconnected acyclic graphs are called forests . Watch as we transform a graph with cycles into an acyclic forest by removing specific edges. 5. Why Cycles Matter in Graph Theory Cycles are fundamental to understanding graph properties and solving real-world problems: Network Analysis: Cycles represent redundant paths and backup routes in networks Circuit Design: Detecting cycles prevents infinite loops in electronic circuits

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