CSE390D Notes for Monday, 12/2/24
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definition:
A graph G = (V, E) consists of V, a nonempty set of vertices (or
nodes) and E, a set of edges. Each edge has either one or two
vertices associated with it, called its endpoints. An edge is set
to connect its endpoints.
adjacent:
2 vertices adjacent in undirected graph if e connects them {u, v}
edge is said to be incident
incident: edge is incident to each vertex
deg(v) = # of edges incident to it (special case for self-loops--2)
handshaking theorem:
what is sum(v in V) deg(v)?
= 2 * |E|
example:
what is true of vertices with odd degree?
must be an even # of them
directed graphs: edges are (u, v) ordered
u is adjacent to v
v is adjacent from u
in-degree = deg^-(v) = # of edges ending in v
out-degree = deg^+(v) = # of edges starting with v
sum(v in V) deg^-(v) = sum(v in V) deg^+(v) = |E|
bipartite graph: vertices can be partioned into two disjoint sets
such that all edges connect a vertex in one set to a vertex in the
other set
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special graphs:
Kn is complete graph on n vertices (all possible connections, but
no self loops)
Kn,m is complete graph on n and m vertices (bipartite graph with
all possible connections between two sets of vertices)
Cn is a cycle of length n (same starting and ending vertex)
subgraph of G = (V, E)
H = (W, F), W subset of V, F subset of E
representation
adjacency list
adjacency matrix
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connectivity
path from v1 to vn is sequence of edges
(v1, v2, v3, ..., vn) where (vi, vi+1) is an edge
simple path if edges are unique
circuit: starting and ending vertex the same
simple circuit: no repeated edges
graph is connected if there is a path between every pair of
vertices
Euler circuit: simple circuit including all edges (exactly once)
Euler path: simple path including all edges (exactly once)
properties of graph with Euler circuit?
all deg are even
Hamiltonian path: simple path with every vertex once
Hamiltonian circuit: simple circuit with every vertex once (except
first/last)
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A Tree is:
connected, acyclic graph
Assumes undirected, nonempty.
Goldilocks point between disconnected and cyclic:
add an edge, you get a cycle
remove an edge, it's disconnected
equivalent definition: unique path between any two vertices
rooted tree: one node identified as root
think of the tree put together with paper clips for edges, pick up the
tree by the root
children, grandchildren, siblings, etc
m-ary tree is a rooted tree where each vertex has at most m children
we're used to 2-ary trees
we're also used to ordered trees
leaf: a vertex of a rooted tree that has no children
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is it true that every tree has a leaf? yes
prove by contradiction
let T be a rooted tree that has no leaf
T has n vertices
start at the root
go to a child
go to a child
go to a child
if it ends, we've got a leaf, so it can't end
do that n times
end up visiting n+1 vertices
pigeon hole principle: 2 vertices the same, cycle, contradiction
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important property of trees:
|V| = |E| + 1
in fact, any 2 of (connected, acyclic, |V| = |E| + 1) means tree
how would we prove this?
induction on vertices
P(1): only one tree of one vertex, it has no edges, 1 = 0 + 1
assume P(k) for some k in Z+, prove P(k+1)
Prove that P(k+1) holds
P(k): vertices = edges + 1 for trees of k vertices
P(k + 1): vertices = edges + 1 for trees of k+1 vertices
Let T be a tree of k+1 vertices
it must have a leaf L
let T' = T - (leaf/edge)
T' is a tree of k vertices
therefore, vertices = edges + 1 in T' (vertices = k, edges = k-1)
but T has 1 more vertex and 1 more edge (vertices=k+1, edges=k)
therefore the property holds of T
this completes the proof
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graph isomorphism: are two graphs the same?
planarity
Kuratowski's theorem
A graph is nonplanar if it contains a subgraph of K3,3 or K5
graph coloring
coloring is assignment of colors to vertices, no adjacent pair
assigned the same color
chromatic number of a graph is least number of colors needed for a
coloring of this graph
four color theorem: chromatic number of a planar graph is no more
than 4
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Stuart Reges
Last modified: Mon Dec 2 14:09:59 PST 2024