Cycle (graph theory)

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Cycle (graph theory)

In graph theory, a cycle is a path of edges and vertices wherein a vertex is reachable from itself by traversing the edges in a sequence where the first and last vertices are the same, and with no other repetition of vertices and edges. Cycles are fundamental concepts in graph theory and have applications in various fields such as computer science, network analysis, and mathematics.

Definition

A cycle in a graph \(G = (V, E)\), where \(V\) represents the set of vertices and \(E\) represents the set of edges, is a sequence of vertices \(v_1, v_2, ..., v_k\) with a sequence of edges \((v_1, v_2), (v_2, v_3), ..., (v_{k-1}, v_k), (v_k, v_1)\) such that all \(v_i\) are distinct for \(1 \leq i < k\), and each edge is included in \(E\). The cycle is often denoted as \(C_k\) indicating a cycle of length \(k\), where \(k\) is the number of edges (or vertices) in the cycle.

Types of Cycles

There are several types of cycles in graph theory, including:

- Simple Cycles: A cycle that does not intersect itself and visits each vertex exactly once, except for the starting/ending vertex. - Hamiltonian Cycles: A cycle that visits each vertex exactly once. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. - Eulerian Cycles: A cycle that visits every edge exactly once. A graph that contains an Eulerian cycle is called an Eulerian graph.

Properties

- A graph is bipartite if and only if it does not contain any odd-length cycles. - The presence of cycles in a graph affects its planarity, connectivity, and coloring properties. - Detecting cycles is a fundamental problem in graph algorithms, with applications in detecting deadlocks in concurrent systems and finding feedback loops in networks.

Cycle Detection

Cycle detection algorithms are used to find cycles within graphs. Common algorithms include Depth-First Search (DFS) for undirected graphs and algorithms based on Topological Sorting for directed graphs without cycles (acyclic graphs).

Applications

Cycles in graphs have numerous applications across different fields: - In computer networks, cycles can represent loops in routing paths. - In bioinformatics, cycles in graphs can represent feedback loops and pathways in biological networks. - In operational research, cycles in graphs can model cyclic processes in manufacturing and logistics.

See Also

- Graph Theory - Vertex (graph theory) - Edge (graph theory) - Hamiltonian Path - Eulerian Path