Graph problems are so ubiquitous that its unimaginable to have a contest set without multiple graph based problems. To be successful, you must become comfortable working with graphs, representing them, transforming them, and executing well known algorithms on them.
What is a Graph
Graph Properties
Directionality
weightedness
Flow
Bipartite
DAGs
Trees
Graph Representation
Adjacency Matrix
Adjacency List
Implied Graphs
In and Out Nodes
Standard Graph Algorithms
Topological Sort
Breadth First Search
Depth First Search
Minimum Spanning Tree
Distance Algorithms
Dijkstra
Bellman Ford
Floyd-Warshall
Strongly Connected Components
Bi-Connected Components
Flow and Matching Algorithms
Max Flow/Min Cut
Min Cost Max Flow
Bipartite Matching/Vertex Cover
Min Cost Matching
State Minimization