Depth-First Search Algorithm in Graph Theory

In graph theory, Depth-First Search (DFS) is a fundamental algorithm frequently employed to address reliability issues in power systems. Beyond this application, DFS proves valuable for solving various network problems including shortest path identification, connectivity analysis, and cycle detection. The core principle of DFS involves initiating traversal from a starting node and progressively exploring deeper along each branch before backtracking. This systematic approach ensures complete exploration of all graph nodes while identifying all possible paths between source and target nodes. From an implementation perspective, DFS typically utilizes either recursion or an explicit stack data structure to manage node visitation states. The algorithm maintains three states for nodes: unvisited, visiting, and visited. Key operations include pushing adjacent nodes onto the stack/recursion chain and marking nodes as visited to prevent redundant processing. This method guarantees O(V+E) time complexity for graphs with V vertices and E edges when using adjacency lists. The backtracking mechanism enables comprehensive path exploration, making DFS particularly suitable for topological sorting, strongly connected components detection, and maze-solving scenarios in network analysis.