Graph Theory
Graph theory is the study of the
properties of graph structures. It
provides us with a language with
which to talk about graphs.
Degree
• The degree of a vertex is the number of
edges incident upon it.
• The sum of the vertex degrees in any
undirected graph is even (twice the
number of edges).
• Every graph contains an even number of
odd-degree vertices.
Connectivity
• A graph is connected if there is an
undirected path between every pair of
vertices.
• The existence of a spanning tree is
sufficient to prove connectivity.
• The vertex (edge) connectivity is the
smallest number of vertices (edges) which
must be deleted to disconnect the graph.
Some terms
• articulation vertex  biconnected
• bridge  edge-biconnected
• Testing for articulation vertices or bridges
is easy via brute force.
Cycles
• Eulerian cycle (path): a tour which visits
every edge of the graph exactly once.
• Actually, it is a circuit, not a cycle, because
it may visit vertices more than once.
• A mailman’s route is ideally an Eulerian
cycle, so he can visit every street (edge) in
the neighborhood once before returning
home.
• An undirected graph contains an Eulerian
cycle if it is connected and every vertex is
of even degree.
• A Hamiltonian cycle is a tour which visits
every vertex of the graph exactly once.
• The traveling salesman problem asks for
the shortest such tour on a weighted graph.
Planer Graph
• Euler’s formula:
n-m+f=2
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n:# of vertices
m:# of edges
f: # of faces
Trees: m=n-1, f=1
Cubes: n=8, m=12, f=6
MST
• Kruskal’s algorithm:
starting from a minimal edge
• Prim’s algorithm:
starting from a given vertex
– How about maximum spanning tree
– and Minimum Product spanning tree
Kruskal’s Algorithm
• Algorithm Kruskal(G)
• Input：G=(V, E)為無向加權圖(undirected weighted graph)，
其中V={v0,…,vn-1}
• Output：G的最小含括樹(minimum spanning tree, MST)
• T← //T為MST，一開始設為空集合
• while T包含少於n-1個邊 do
• 選出邊(u, v)，其中(u, v)E，且(u, v)的加權(weight)最小
• E←E-(u, v)
• if ( (u, v)加入T中形成循環(cycle) ) then 將(u, v)丟棄
• else T←T(u, v)
• return T
Kruskal’s Algorithm -Construct MST
Prim’s algorithm
• Algorithm Prim(G)
• Input：G=(V, E)為無向加權圖(undirected weighted graph)，其
中V={v0,…,vn-1}
• Output：G的最小含括樹(minimum spanning tree, MST)
• T← //T為MST，一開始設為空集合
• X←{vx} //隨意選擇一個頂點vx加入集合X中
• while T包含少於n-1個邊 do
• 選出(u, v)E，其中uX且vV-X，且(u, v)的加權(weight)最小
• T←T(u, v)
• X←X{v}
• return T
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One-to-all shorted path
• Dijkstra演算法: Dijkstra演算法屬於求取單一來源(source)至所有目
標(destination)頂點的一至多(one-to-all)最短路徑演算法
All-pair shortest path
• Floyd-Warshall的所有頂點對最短路徑(all-pair shortest path)演
算法：