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notes-and-examples/2026.07.21/depth-first-search.md

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+## Traversal introduction
+
+Many algorithms that involve graphs must involve some way to traverse the elements of a graph. The two simplest ways of traversal are depth-first search (DFS) and [breadth-first search (BFS)](https://en.wikipedia.org/wiki/Breadth-first_search). The major difference here is the *order* in which nodes are traversed.
+
+If we start from vertex 0 in the tree, in what order would you expect depth-first search to traverse the nodes? (There are multiple correct answers!)
+
+```
+        0
+       / \
+      1   2
+     / \
+    3   4
+```
+
+Note that a single traversal step checks for already-visited nodes. So, if the path is `0 -> 1 -> 3`, the path cannot become `0 -> 1 -> 3 -> 1`.
+
+What about starting from vertex 0 in this graph?
+
+```
+    0 --- 1
+    |     |
+    3 --- 2
+```
+
+What about this one?
+
+```
+        0
+       / \
+      1   5
+     / \   \
+    2   3   6
+     \ /   /
+      4   7
+       \ /
+        8
+```
+
+## Formalizing the algorithm
+
+Based on these examples, can we create a formal algorithm that takes a starting node, producing a valid traversal path for all three graphs?
+
+An example for this graph:
+
+```
+    0 --- 1
+    |     |
+    3 --- 2
+```
+
+- Arguments: starting node `0`
+- Expected output: