Graphs 1.3 - Spanning Trees A spanning tree of a connected...

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Unformatted text preview: Spanning Trees A spanning tree of a connected graph is a spanning subgraph that is a tree A spanning tree is not unique unless the graph is a tree Spanning trees have applications to the design of communication networks Graph A spanning forest of a graph is a spanning subgraph that is a forest Spanning tree CSE 2011 Prof. J. Elder - 11 - Last Updated: 4/1/10 10:16 AM Reachability in Directed Graphs A node w is reachable from v if there is a directed path originating at v and terminating at w. E is reachable from B B is not reachable from E E D C F A CSE 2011 Prof. J. Elder B - 12 - Last Updated: 4/1/10 10:16 AM Properties Property 1 Notation v deg(v) |V| number of edges deg(v) Proof: each edge is counted twice number of vertices |E| = 2|E| degree of vertex v Property 2 In an undirected graph with no self-loops and no multiple edges |E| |V| (|V| Example |V| = 4 |E| = 6 deg(v) = 3 1)/2 Proof: each vertex has degree at most (|V| 1) Q: What is the bound for a digraph? A: E V (V CSE 2011 Prof. J. Elder 1) - 13 - Last Updated: 4/1/10 10:16 AM Main Methods of the (Undirected) Graph ADT Vertices and edges Update methods are positions insertVertex(o): insert a vertex storing element o store elements insertEdge(v, w, o): insert an edge (v,w) storing element o Accessor methods endVertices(e): an array of the two endvertices of e removeVertex(v): remove vertex v (and its incident edges) opposite(v, e): the vertex opposite to v on e removeEdge(e): remove edge e Iterator methods areAdjacent(v, w): true iff v and w are adjacent replace(v, x): replace element at vertex v with x replace(e, x): replace element at edge e with x CSE 2011 Prof. J. Elder - 14 - incidentEdges(v): edges incident to v vertices(): all vertices in the graph edges(): all edges in the graph Last Updated: 4/1/10 10:16 AM Directed Graph ADT Additional methods: isDirected(e): return true if e is a directed edge insertDirectedEdge(v, w, o): insert and return a new directed edge with origin v and destination w, storing element o CSE 2011 Prof. J. Elder - 15 - Last Updated: 4/1/10 10:16 AM ...
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This note was uploaded on 02/14/2012 for the course CSE 2011Z taught by Professor Elder during the Fall '11 term at York University.

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