Graph Theory/k-Connected Graphs

Definition of connectedness Let u and v be a vertex of graph . If there is a path in , then we say that and are connected. If there is a path for every pair of vertices and in , then we say that is connected or connected graph. Edge Connectivity The minimum number of edges lambda() whose deletion from a graph disconnects , 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. Vertex Connectivity The minimum number of vertices kappa() whose deletion from a graph disconnects it. Let lambda() be the edge connectivity of a graph and delta() its minimum degree, then for any graph, kappa() ≤ lambda() ≤ delta() k-connected Graph k-edge-connected Graph 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. k-vertex-connected Graph 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. Menger's Theorem edge connectivity The size of the minimum edge cut for and (the minimum number of edges whose removal disconnects and ) is equal to the maximum number of pairwise edge-disjoint paths from to vertex connectivity The size of the minimum vertex cut for and (the minimum number of vertices whose removal disconnects and ) is equal to the maximum number of pairwise vertex-disjoint paths from to ( 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. ) max-flow( maximum flow ) min-cut( minimum cut ) Theorem The maximum flow between vertices and in a graph is exactly the weight of the smallest set of edges to disconnect with and in different components. maximum flow : The maximum flow between vertices and in a graph minimum cut : the smallest set of edges to disconnect with and in different components.