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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 = 2E degree of vertex v Property 2
In an undirected graph with no
selfloops 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.
 Fall '11
 Elder
 Data Structures

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