docs(docs): add ProblemBadges component, layout slot, and enrich pages with topics/video frontmatter, plus guide placeholders and topic‑landing map.

apps/docs/content/(04-relational)/graph-bfs/127-word-ladder.mdx

@@ -2,13 +2,16 @@
 title: '127. Word Ladder'
 description: 'A transformation sequence from word beginWord to word endWord using a dictionary wordList is a sequence of words beginWord -> s1 -> s2 -> ... -> sk such that:'
 icon: dot
+topics:
+  - { name: "Hash Table", slug: "hash-table" }
+  - { name: "String", slug: "string" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Bidirectional Search", slug: "bidirectional-search" }
 sidebar:
   label: 'Word Ladder'
   badge: 'Hard'
 ---
 
-<Badge variant="accent">Hash Table</Badge>
-
 ### Example 1:
 - Input: `beginWord = "hit", endWord = "cog", wordList = ["hot","dot","dog","lot","log","cog"]`
 - Output: `5`

apps/docs/content/(04-relational)/graph-bfs/994-rotting-oranges.mdx

@@ -2,13 +2,15 @@
 title: '994. Rotting Oranges'
 description: 'You are given an m x n grid where each cell can have one of three values:'
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Matrix", slug: "matrix" }
 sidebar:
   label: 'Rotting Oranges'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `grid = [[2,1,1],[1,1,0],[0,1,1]]`
 - Output: `4`

apps/docs/content/(04-relational)/graph-dfs/133-clone-graph.mdx

@@ -2,13 +2,16 @@
 title: '133. Clone Graph'
 description: Given a reference of a node in a connected undirected graph
 icon: dot
+topics:
+  - { name: "Hash Table", slug: "hash-table" }
+  - { name: "Depth-First Search", slug: "depth-first-search" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Graph Theory", slug: "graph" }
 sidebar:
   label: 'Clone Graph'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Hash Table</Badge>
-
 ### Example 1:
 - Input: `adjList = [[2,4],[1,3],[2,4],[1,3]]`
 - Output: `[[2,4],[1,3],[2,4],[1,3]]`

apps/docs/content/(04-relational)/graph-dfs/200-number-of-islands.mdx

@@ -2,13 +2,17 @@
 title: '200. Number of Islands'
 description: Given an m x n 2D binary grid grid which represents a map of '1's (land) and '0's (water), return the number of islands
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Depth-First Search", slug: "depth-first-search" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Union-Find", slug: "union-find" }
+  - { name: "Matrix", slug: "matrix" }
 sidebar:
   label: 'Number of Islands'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `grid = [ ["1","1","1","1","0"], ["1","1","0","1","0"], ["1","1","0","0","0"], ["0","0","0","0","0"] ]`
 - Output: `1`

apps/docs/content/(04-relational)/graph-dfs/417-pacific-atlantic-water-flow.mdx

@@ -2,13 +2,16 @@
 title: '417. Pacific Atlantic Water Flow'
 description: There is an m x n rectangular island that borders both the Pacific Ocean and Atlantic Ocean. The Pacific Ocean touches the island's left and top edges, and the Atlantic Ocean touches the island's right and bottom edges
 icon: dot
+topics:
+  - { name: "Array", slug: "array" }
+  - { name: "Depth-First Search", slug: "depth-first-search" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Matrix", slug: "matrix" }
 sidebar:
   label: 'Pacific Atlantic Water Flow'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Array</Badge>
-
 ### Example 1:
 - Input: `heights = [[1,2,2,3,5],[3,2,3,4,4],[2,4,5,3,1],[6,7,1,4,5],[5,1,1,2,4]]`
 - Output: `[[0,4],[1,3],[1,4],[2,2],[3,0],[3,1],[4,0]]`

apps/docs/content/(04-relational)/topological-sort/207-course-schedule.mdx

@@ -2,13 +2,17 @@
 title: '207. Course Schedule'
 description: There are a total of numCourses courses you have to take, labeled from 0 to numCourses - 1. You are given an array prerequisites where prerequisites[i] = [ai, bi] indicates that you must take course bi first if you want to take course ai
 icon: dot
+topics:
+  - { name: "Depth-First Search", slug: "depth-first-search" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Graph Theory", slug: "graph" }
+  - { name: "Topological Sort", slug: "topological-sort" }
+  - { name: "Directed Acyclic Graph", slug: "directed-acyclic-graph" }
 sidebar:
   label: 'Course Schedule'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Depth-First Search</Badge>
-
 ### Example 1:
 - Input: `numCourses = 2, prerequisites = [[1,0]]`
 - Output: `true`

apps/docs/content/(04-relational)/topological-sort/210-course-schedule-ii.mdx

@@ -2,13 +2,16 @@
 title: '210. Course Schedule II'
 description: There are a total of numCourses courses you have to take, labeled from 0 to numCourses - 1. You are given an array prerequisites where prerequisites[i] = [ai, bi] indicates that you must take course bi first if you want to take course ai
 icon: dot
+topics:
+  - { name: "Depth-First Search", slug: "depth-first-search" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Graph Theory", slug: "graph" }
+  - { name: "Topological Sort", slug: "topological-sort" }
 sidebar:
   label: 'Course Schedule II'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Depth-First Search</Badge>
-
 ### Example 1:
 - Input: `numCourses = 2, prerequisites = [[1,0]]`
 - Output: `[0,1]`

apps/docs/content/(04-relational)/union-find/323-number-of-connected-components-in-an-undirected-graph.mdx

@@ -2,13 +2,16 @@
 title: '323. Number of Connected Components in an Undirected Graph'
 description: You have a graph of n nodes. You are given an integer n and an array edges where edges[i] = [ai, bi] indicates that there is an edge between ai and bi in the graph
 icon: dot
+topics:
+  - { name: "Depth-First Search", slug: "depth-first-search" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Union-Find", slug: "union-find" }
+  - { name: "Graph Theory", slug: "graph" }
 sidebar:
   label: 'Number of Connected Components in an Undirected Graph'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Depth-First Search</Badge>
-
 ### Example 1:
 - Input: `n = 5, edges = [[0,1],[1,2],[3,4]]`
 - Output: `2`

apps/docs/content/(04-relational)/union-find/684-redundant-connection.mdx

@@ -2,13 +2,16 @@
 title: '684. Redundant Connection'
 description: In this problem, a tree is an undirected graph that is connected and has no cycles
 icon: dot
+topics:
+  - { name: "Depth-First Search", slug: "depth-first-search" }
+  - { name: "Breadth-First Search", slug: "breadth-first-search" }
+  - { name: "Union-Find", slug: "union-find" }
+  - { name: "Graph Theory", slug: "graph" }
 sidebar:
   label: 'Redundant Connection'
   badge: 'Medium'
 ---
 
-<Badge variant="accent">Depth-First Search</Badge>
-
 ### Example 1:
 - Input: `edges = [[1,2],[1,3],[2,3]]`
 - Output: `[2,3]`