What is a cut in a graph?

A cut in a graph is a partition of the vertices of the graph into two disjoint sets. A cut is defined by the edges that cross between the two sets. Cuts are used in many applications, including network flow analysis and clustering.

What is the shortest path problem?

The shortest path problem is the problem of finding the shortest path between two vertices in a graph. There are several algorithms for solving the shortest path problem, including Dijkstra’s algorithm, Bellman-Ford algorithm, and Floyd-Warshall algorithm.

What is the maximum flow problem?

The maximum flow problem is the problem of finding the maximum flow that can be sent through a network of pipes, where each pipe has a capacity. There are several algorithms for solving the maximum flow problem, including the Ford-Fulkerson algorithm and the Edmonds-Karp algorithm.

What is the minimum cut problem?

The minimum cut problem is the problem of finding the cut with the smallest capacity that separates two vertices in a graph. There are several algorithms for solving the minimum cut problem, including the Stoer-Wagner algorithm and the Karger’s algorithm.

What is graph coloring?

Graph coloring is the problem of assigning colors to the vertices of a graph such that no two adjacent vertices have the same color. Graph coloring has many applications, including scheduling, map coloring, and register allocation in compilers.

What is a strongly connected component (SCC) in a directed graph?

A strongly connected component (SCC) in a directed graph is a subgraph where there is a directed path between every pair of vertices in the subgraph. SCCs are used in many applications, including control flow analysis, network analysis, and database design.

What is the traveling salesman problem?

The traveling salesman problem (TSP) is the problem of finding the shortest possible route that visits every vertex exactly once and returns to the starting vertex in a weighted graph. The TSP is a well-known NP-hard problem, and there are several algorithms for approximating the optimal solution, such as the nearest neighbor algorithm and the Christofides algorithm.

What is a planar graph?

A planar graph is a graph that can be embedded in a plane without any edges crossing. Planar graphs have many interesting properties, and they are used in many applications, including circuit design, graph theory, and computer graphics.

What is a Eulerian path or circuit?

A Eulerian path is a path that visits every edge of a graph exactly once. A Eulerian circuit is a circuit that visits every edge of a graph exactly once and returns to the starting vertex. Eulerian paths and circuits are used in many applications, including transportation networks, DNA sequencing, and computer networking.

What is a Hamiltonian path or circuit?

A Hamiltonian path is a path that visits every vertex of a graph exactly once. A Hamiltonian circuit is a circuit that visits every vertex of a graph exactly once and returns to the starting vertex. Hamiltonian paths and circuits are used in many applications, including routing optimization, scheduling, and optimization of DNA sequencing.