VIEWS: 44 PAGES: 19 CATEGORY: Education POSTED ON: 10/27/2011 Public Domain
Paths & Circuits Paths & Cycles Euler Paths • An Euler path in a graph is a simple path containing every edges of that graph. Euler Circuits • An Euler circuit in a graph is a simple circuit containing every edges of that graph. Euler Paths & Circuits • An Euler circuit in a graph is a circuit that traverses all the edges in the graph once. • An Euler path in a graph is a path that traverses all the edges in the graph once. • An undirected multi-graph has an Euler circuit if and only if it is connected and has all vertices of even degree. • An undirected multi-graph has an Euler path, but not Euler circuit, if and only if it is connected and has exactly two vertices of odd degree. Condition of Euler Circuits • A connected multi-graph with at least two vertices has an Euler circuit ↔ each of its vertices has even degree. • Necessary condition: G has an Euler Circuit → each of its vertices must have even degree. • Sufficient condition: Each of the vertices in G has even degree → G has an Euler Circuit. Find an Euler Circuit • G has an Euler Circuit → each of its vertices must have even degree. • Each of the vertices in G has even degree → G has an Euler Circuit. Find an Euler Circuit Condition of Euler path • A connected multi-graph with at least two vertices has an Euler path ↔ it has exactly 2 vertices with odd degree. Find an Euler Path Euler Circuit Problem ลากเส้ นตามรูปโดยไม่ ซ้า (Euler path) (Euler Circuit) Euler Paths & Circuits Hamilton Paths & Circuits • A Hamilton path in a graph is a simple path that passes through every vertex of the graph exactly once. • For G=(V,E) and V = {v1, v2, …, vn}, the simple circuit v1,v2,…,vn,v0 is a Hamilton circuit if v1,v2,…,vn is a Hamilton path. Hamilton Paths & Circuits • A Hamilton circuit in a graph is a (simple) circuit that visits each vertex in the graph once. • A Hamilton path in a graph is a (simple) path that visits each vertex in the graph once Iconion Puzzle Condition of Hamilton Circuits • No ‘necessary & sufficient’ conditions exist. • Certain properties can be used to show that no Hamilton circuits exist. E.g. degree one vertex. • Both edges incident of a vertex of degree two must be part of any Hamilton circuit. • While constructing a Hamilton circuit, if a vertex has already passes through, all remaining edges of that vertex can be removed from consideration. Some sufficient condition • If G is a simple graph with n vertices (n≥3) such that the degree of every vertex in G is at least n/2, then G has a Hamilton circuit. • If G is a simple graph with n vertices (n≥3) such that deg(u)+deg(v) ≥ n for every pair of non-adjacent vertices u and v in G, then G has a Hamilton circuit. Hamilton Circuit ลากผ่ านจุดทุกจุดๆ ละหนึ่งครั้ง 1 5 10 2 1 2 3 4 3 13 4 5 6 กราฟที่มีวงจร Hamilton 11 7 8 9 12 William Rowan Hamilton (1805–1865) กราฟที่ไม่มีวงจร Hamilton Traveling Salesperson Problem • A salesman is required to visit a number of cities during a trip. Given the distances between cities, in what order should he travel so as to visit every city precisely once and return home, with the minimum mileage traveled ? Input Output