• Edge Weight Function: $w:E\to \R$
• Examples: Functions, Binary relations, Maps, Web links

Graph Problems

1. Matching

Let $G=(V,E)$ be a graph. A matching $M \in G$ is a set of pairwise non-adjacent edges.

• Perfect Matching: Every vertex appear exactly once.
• Weight of Matching = Sum of weights of vertices.
• Problem: Find matching with the maximum weights

Paths

• Two vertices $u, v \in V$ are adjacent if $\{u,v\}\in E$
• TODO
• Problem: Find shortest path
  • Example: Wikipedia Game

Cycles (Undirected Graphs)

• Cycle: $(v1, \dots, v_n)$ is a cycle if $(v_1, \dots, v_{n-1})$ is a path, $v_1 = v_n$ and $\{v_{n-1}, v_n\} \in E$
• Hamiltonian Cycle: Every vertex appear in the cycle exactly once (except for start/end)
• Problem: Traveling Salesman Problem, minimize total weight

Coloring Problem

• Let $C$ be a set of $k$ colors, assign each vertex a color so that adjacent vertices have different colors
• Coloring Function: $f:V\to C$
• Problem: Precoloring Extension: Given a partial coloring, complete it
  • E.g. Sudoku