Graph Theory/k-Connected Graphs

Let u and v be a vertex of graph $$G$$.
 * Definition of connectedness
 * If there is a $$u-v$$ path in $$G$$, then we say that $$u$$ and $$v$$ are connected.
 * If there is a $$u-v$$ path for every pair of vertices $$u$$ and $$v$$ in $$G$$, then we say that $$G$$ is connected or connected graph.

The minimum number of edges lambda($$G$$) whose deletion from a graph $$G$$ disconnects $$G$$, also called the line connectivity. The edge connectivity of a disconnected graph is 0, while that of a connected graph with a graph bridge is 1.
 * Edge Connectivity

The minimum number of vertices kappa($$G$$) whose deletion from a graph $$G$$ disconnects it.
 * Vertex Connectivity

Let lambda($$G$$) be the edge connectivity of a graph $$G$$ and delta($$G$$) its minimum degree, then for any graph, kappa($$G$$) ≤ lambda($$G$$) ≤ delta($$G$$)

A graph has edge connectivity k if k is the size of the smallest subset of edges such that the graph becomes disconnected if you delete them. A graph has vertex connectivity k if k is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. A 1-connected graph is called connected; a 2-connected graph is called biconnected. A 3-connected graph is called triconnected. The size of the minimum edge cut for $$u$$ and $$v$$ (the minimum number of edges whose removal disconnects $$u$$ and $$v$$) is equal to the maximum number of pairwise edge-disjoint paths from $$u$$ to $$v$$ The size of the minimum vertex cut for $$u$$ and $$v$$ (the minimum number of vertices whose removal disconnects $$u$$ and $$v$$) is equal to the maximum number of pairwise vertex-disjoint paths from $$u$$ to $$v$$ ( An edge cut is a set of edges whose removal disconnects the graph, and similarly a vertex cut or separating set is a set of vertices whose removal disconnects the graph. )
 * k-connected Graph
 * k-edge-connected Graph
 * k-vertex-connected Graph
 * Menger's Theorem
 * edge connectivity
 * vertex connectivity

The maximum flow between vertices $$u$$ and $$v$$ in a graph $$G$$ is exactly the weight of the smallest set of edges to disconnect $$G$$ with $$u$$ and $$v$$ in different components.
 * max-flow( maximum flow ) min-cut( minimum cut ) Theorem
 * maximum flow : The maximum flow between vertices $$u$$ and $$v$$ in a graph $$G$$
 * minimum cut : the smallest set of edges to disconnect $$G$$ with $$u$$ and $$v$$ in different components.