Graphs

Depth-First Search (Recursive)

Visit a start vertex, then recurse into its first unvisited neighbour all the way down before backtracking. A visited set prevents revisiting, and neighbour insertion order fixes the visit sequence.

Algorithm

On the canonical 6-vertex graph from graph-adjacency-list, starting at vertex 1, the deterministic visit order is [1, 2, 4, 3, 5, 6]. Calls unwind 6 -> 5 -> 4 -> 3 -> 2 -> 1 after all vertices are visited.

recursive descent Follow one branch to its end, then unwind and try the next neighbour.

Visual walkthrough

DFS follows the first unvisited neighbour as far as it can, then unwinds and tries the next branch.

Step 1 - First branch

Starting from 1, insertion order sends DFS to 2, then 4, then 3.

Early DFS descent: 1 -> 2 -> 4 -> 3.1#12#23#44#356

Step 2 - Full DFS order

The final order is [1, 2, 4, 3, 5, 6], then calls unwind 6 -> 5 -> 4 -> 3 -> 2 -> 1.

DFS order and final recursive tail.1#12#23#44#35#56#6

Basic Implementation

basic.kt 
fun main() {
	val adj = HashMap<Int, List<Int>>()
	adj[1] = listOf(2, 3)
	adj[2] = listOf(1, 4)
	adj[3] = listOf(1, 4)
	adj[4] = listOf(2, 3, 5)
	adj[5] = listOf(4, 6)
	adj[6] = listOf(5)
	val visited = HashSet<Int>()
	val order = mutableListOf<Int>()
	fun dfs(v: Int) {
		visited.add(v)
		order.add(v)
		for (nb in adj[v]!!) {
			if (!visited.contains(nb)) {
				dfs(nb)
			}
		}
	}
	dfs(1)
	println(order.joinToString(prefix = "[", postfix = "]"))
}

Complexity

  • Time: O(V + E)
  • Space: O(V) recursion depth

Implementation notes

  • Kotlin: a local recursive dfs closes over the shared visited set and order list.
  • The replay shows the current vertex, the visited set, and the running visit order after each entry, matching the lesson spec.