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={v 0,…,v n-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={v 0,…,v n-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←{v x } // 隨意選擇一個頂點 v x 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

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：