graphPropsSmry.txt

(Latest Revision: Wed Mar 7, 2018 ) graphPropsSmry.txt
graphPropsSmry.txt


Summary of Basic Facts about Undirected Graphs and Trees

Proposition 1: 
Let G be an undirected graph.
   G is a tree if and only if 
   G is connected and G has no cycles. 

Proposition 2: 
Let G be a connected undirected graph.
G has a subgraph that is a spanning tree of G.

Proposition 3: 
Let G  be a connected undirected graph with n nodes 
G has at least n-1 edges.

Corollary to Proposition 3: 
Let G be a connected undirected graph with n nodes 
and n-1 edges. G is a tree.

Proposition 4: 
Let G be an undirected graph with n nodes and more 
than n-1 edges.  G contains a cycle.

Corollary to Proposition 4:
Let G be a tree with n nodes.
G has exactly n-1 edges.  (no more and no less)

Definition 5:
A set T of edges of an undirected graph G is promising 
if it can be extended to form the edge set 
of a minimal cost spanning tree of G.

Definition 6:
Let B be a non-empty proper subset of the node set V 
of an undirected graph G. An edge e leaves B
if one end of e is in B and the other end is in V-B.

Lemma 6.3.1:
Let G be a connected undirected graph, each edge having a 
positive length. Let B be a non-empty proper subset 
of the node set V of G.  Let T be a promising subset 
of the edge set E of G, such that no edge in T leaves B.
Let v be a shortest edge that leaves B.  Then the union 
of T with the singleton {v} is promising.

Theorem 6.3.2:
If G is a connected undirected graph, then Kruskal's 
algorithm finds a minimum cost spanning tree of G.
(In other words, Kruskal's algorithm works.)