From f117ec52e505a4825b34a16880e5cf44e032f8e8 Mon Sep 17 00:00:00 2001
From: bt3gl <138340846+bt3gl-cryptographer@users.noreply.github.com>
Date: Tue, 8 Aug 2023 16:12:42 -0700
Subject: [PATCH] Update README.md

---
 graphs/README.md | 83 ++++++++++++++++++++++++++++++++++++++++++++++--
 1 file changed, 80 insertions(+), 3 deletions(-)

diff --git a/graphs/README.md b/graphs/README.md
index 30dbe16..f583d17 100644
--- a/graphs/README.md
+++ b/graphs/README.md
@@ -3,10 +3,18 @@
 <br>
 
 * graph is a non-linear data structure consisting of vertices and edges.
+
+* graphs can be represented by adjacent matrices, adjacent lists, and hash table of hash tables.
+  
 * in **undirected graphs**, the edges between any two vertices do not have a direction, indicating a two-way relationship.
+  
 * in **directed graphs**, the edges between any two vertices are directional.
+  
 * in **weighted graphs**, each edge has an associated weight. if the sum of the weights of all edges of a cycle is a negative values, it's a negative weight cycle.
-* the **degree of a vertex** is the number of edges connecting the vertex. in directed, graphs, if the **in-dregree** of a vertex is d, there are **d** directional edges incident to the vertex (and similarly, **out-degree** from the vertex).
+  
+* the **degree of a vertex** is the number of edges connecting the vertex. in directed, graphs, if the **in-dregree** of a vertex is `d`, there are **d** directional edges incident to the vertex (and similarly, **out-degree** from the vertex).
+
+* with `|V|` the number of vertices and `|E|` is the number of edges, search in a graph (either bfs of dfs) is `O(|V| + |E|)`.
 
 <br>
 
@@ -16,6 +24,75 @@
 
 <br>
 
-* check **[../trees/#breath-first-search](https://github.com/go-outside-labs/master-python-with-algorithms-py/tree/master/trees#breath-first-search)**
-* and **[../trees/#depth-first-search](https://github.com/go-outside-labs/master-python-with-algorithms-py/tree/master/trees#depth-first-search)**
+#### breath first search
+
+<br>
+
+* check **[../trees/#breath-first-search](https://github.com/go-outside-labs/master-algorithms-py/blob/master/trees/README.md#tree-traversal-depth-first-search)**
+
+<br>
+
+```python
+def bfs(matrix):
+
+  if not matrix:
+    return []
+
+  rows, cols = len(matrix), len(matrix[0])
+  visited = set()
+  directions = ((0, 1), (0, -1), (1, 0), (-1, 0))
+
+  def traverse(i, j):
+    queue = deque([(i, j)])
+    while queue:
+      curr_i, curr_j = queue.popleft()
+      if (curr_i, curr_j) not in visited:
+        visited.add((curr_i, curr_j))
+
+        for direction in directions:
+          next_i, next_j = curr_i + direction[0], curr_j + direction[1]
+          if 0 <= next_i < rows and 0 <= next_j < cols:
+
+            queue.append((next_i, next_j))
+
+  for i in range(rows):
+    for j in range(cols):
+      traverse(i, j)
+```
+
+<br>
+
+----
+
+#### depth first search
+
+<br>
+
+* and **[../trees/#depth-first-search](https://github.com/go-outside-labs/master-algorithms-py/blob/master/trees/README.md#tree-traversal-breath-first-search-level-order)**
+
+<br>
+
+```python
+def dfs(matrix):
+  if not matrix:
+    return []
+
+  rows, cols = len(matrix), len(matrix[0])
+  visited = set()
+  directions = ((0, 1), (0, -1), (1, 0), (-1, 0))
+
+  def traverse(i, j):
+    if (i, j) in visited:
+      return
+
+    visited.add((i, j))
+    for direction in directions:
+      next_i, next_j = i + direction[0], j + direction[1]
+      if 0 <= next_i < rows and 0 <= next_j < cols:
+        traverse(next_i, next_j)
+
+  for i in range(rows):
+    for j in range(cols):
+      traverse(i, j)
+```