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

• 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 uX 且 vV-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：