Unformatted text preview: COMP271H COMP271H
Tutorial 5 Biconnected graph Biconnected component Biconnected graph Biconnected Definition: A connected graph that is NOT broken by NOT removal of a single vertex (and its edges) single from the graph from It implies: At least two vertices removal (if It any) are needed to disconnect the graph any) A biconnected graph contains NO articulation biconnected point (AP) point Another definition: Another Examples: Biconnected graph Examples: Not biconnected graph Why not?? Reason: It is NOT connected It is biconnected Why? Reason: It is connected and contains no AP (formal definition) Examples: Biconnected graph Examples: Trees (>2 nodes) are NOT biconnected Why? Reason: It contains AP Complete graph is biconnected Why? Reason: It is connected and removal of any single vertex still ensure connectivity Biconnected components Biconnected A biconnected component is defined as biconnected the maximal biconnected subgraph maximal It does not contain any larger subgraph It larger inside it If G’ is a biconnected component, then any If subgraph (not including G’) is NOT biconnected biconnected Biconnected components Biconnected Not a biconnected component Not a biconnected component A biconnected component Application Application In a network design, we want to have a In robust network robust Network with AP is NOT robust Reason: If that AP fails, the network is disconnected disconnected Goal: Find a network of N nodes, which contains no AP and number of edges is no minimal minimal Solution: Biconnected component of N nodes Today’s tutorial Today’s (Unrelated to application) Input: A connected graph Output: Identification of connected components Algorithm design (Observations) Algorithm Key observations: Two different biconnected components Two CANNOT have a common edge (but they can CANNOT have common vertex) have That common vertex MUST BE an AP (Why?) That MUST
• Reason: Input graph is connected, the only thing makes it NOT biconnected is the existence of AP makes These APs “separate” different biconnected These components components Idea: See AP, split the AP into two vertices DFSbased algorithm DFSbased Should be similar to finding AP Idea: Algorithm workflow Algorithm Possible way to modify Possible pseudocode of ArtPt(v) Initial condition…. Create a new empty stack Instead of output v, we create a new stack, Instead push (v,w) into the stack push push (v,w) into the stack push If we detect AP…. Otherwise…. ...
View
Full Document
 Spring '10
 may
 English, Graph Theory, Planar graph, complete graph, Graph connectivity, Kvertexconnected graph

Click to edit the document details