그래프 이론 그래프이론 그래프 알고리즘 4색 문제

그래프 란? Node (virtices) 점 Edges 선분 Directed graph (digraph) Weighted graph 그래프는 많은 것을 표현할수 있다. 도로망, 전산망, 분자, 인간관계, 사회조직, 데이터 구조, 생물유전자 관계, 고고학에서 유물연구….

그래프의 표현 list edges and end point vertices Vertices A, B, C, g, w, e Edges Ag, Aw, Ae, Bg, Bw, Be, Cg, Cw, Ce Multiple graph: more than one edges connect given two vertices or loops water gas electricity C A B

그래프의 행렬 표현 (Adjacency matrix)

Incidence Matrix

그래프의 종류 Null graph: 연결안됨 Complete graph: 두점은 반드시 한선분으로 연결됨 Cycle graph: single cycle Path graph: single path Bipartite graph: vertices into two groups ; complete bipartite graph

Platonic graph

Cube graph Petersen graph (generalized) http://mathworld.wolfram.com/GeneralizedPetersenGraph.html

trees Cycle이 없다.

그래프이론의 주문제 Euler path problem (Chinese postman problem) Shortest path problem Minimum spanning tree problem Traveling salesman problem Coloring problem (4색문제) Flow problem Isomorphism problems (Graph matching) Canonical Labeling, subgraph isomorphism and monomorphisms, Maximal common subgraph Graph embedding problem (Planarity)

오일러 길문제 (Euler path problem) 각 선분을 한번 지나고 제자리로 돌아오는 길은? Euler circuit problem http://mathworld.wolfram.com/KoenigsbergBridgeProblem.html

Handshaking Lemma Degree of a node: number of edges ending at a node (no loops) Handshaking Lemma Sum of all degree is twice the number of edges. The sum of vertex-degrees are even. Any graph has an even number of vertices of odd degree

오일라 문제의 일반적인답 그래프는 연결되어 있어야 한다. 그래프가 odd degree의 vertex가 없으면 Euler circuit 이 있다. 그래프가 두개의 odd degree vertex가 있으면 이를 연결하는 Euler path가 있다. 그래프가 4개이상의 odd degree vertex를 가지면 Euler path가 존재하지 않는다.

오일라문제의 해결 만약 Euler circuit이 존재한다면…. Induction 점 하나만 있는 그래프는 만족하는가? Cycle을 하나 찾자. Cycle은 항상 존재하는가? Cycle을 빼면 나머지 그래프는 어떤 조건을 만족하는가?

Fleury's Algorithm Input: A connected graph G each of whose vertices has an even degree. Output: An Eulerian trail C of G. Method: Expand a trail Ci while avoiding bridges in G-Ci, until no other choice remains.

Choose and v0 in V(G) and let C0=v0. Set i:=0. Suppose that the trail C=v0,e1,v1, . . . , ei,vi has already been choosen: At vi choose any edge ei+1 that is not on Ci and that is not a bridge of the graph Gi=G-E(Ci) unless there is no other choice. Define Ci+1=Ci,ei+1,vi+1. Set i:=i+1. If i=|E(G)|, then halt since C=Ci is the desired circuit; else go to 2. 이 알고리즘은 P일까 NP일까? P

Eulerian Type Problems Diagram tracing puzzles L. Poinsot 1809 n개의interconnected point n홀 가능 n짝 불가능

Mazes, Labyrinths http://www.flint.umich.edu/Departments/ITS/crac/maze.form.html Gastron tarry 1895: 이미지나온 교차점으로 가는 길로 가능하면 돌아가지 말라.

Chinese Postman Problem 우체부가 모든 거리의 집에 편지를 전하고 우체국으로 돌아오려한다. 최저 거리의 길을 찾아라. (1962 Meigu Guan 질문) 도시의 눈치우기 (Zurich)

Shortest Path Problem Weighted graph가 주어진경우 두점을 연결하고 weight의 합이 최소인 경로찾기. http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm Application to scheduling.

Spanning tree problem A minimum spanning tree is a tree formed from a subset of the edges in a given undirected graph, with two properties: It spans the graph - it includes every vertex in the graph It is a minimum - the total weight of all the edges is as low as possible http://en2.wikipedia.org/wiki/Minimum_spanning_tree

http://en2.wikipedia.org/wiki/Kruskal%27s_algorithm http://en.wikipedia.org/wiki/Prim's_algorithm

Traveling salesman problem A Hamilitonian cycle: a loop that visits each node once http://mathworld.wolfram.com/HamiltonianCircuit.html NP-Hard http://www.math.princeton.edu/tsp/

Algorithms for tsp http://en.wikipedia.org/wiki/Nearest_neighbour_algorithm http://www.pcug.org.au/~dakin/tsp.htm

Flows: minmax cut theorem http://en.wikipedia.org/wiki/Nearest_neighbour_algorithm

Coloring problem The assignment of labels or colors to the edges or vertices of a graph. The most common types of graph colorings are edge coloring and vertex coloring.

Chromatic number X(G): the minimum number of colors for vertex covering of G Chromatic polynomial PG(k): the number of ways to k color. Chromatic index X’(G): the minimum number of colors for edge-coloring of G

Planarity 오른쪽: planar imbedding, 왼쪽: not planar imbedding 단 선은 굽어도 된다. http://mathworld.wolfram.com/PlanarGraph.html

오일러 공식 G를 연결된 planar 그래프라하자. V: # nodes, E: # edges, F: # regions V – E + F = 2 (주의: 책과 조금 F의 정의 다름) http://www.math.ohio-state.edu/~fiedorow/math655/Euler.html

증명: A: 모든 그래프는 spanning tree에 edge를 붙여서 만든다. B: edge를 하나 붙일때 마다: vertex의 개수는 불변 Face의 개수는 1개는다. 따라서…..

Kuratowski’s theorem A graph is planar if and only if it does not contain a subdivision of K5 or K3,3. K5 는 5개의 node의 complete graph. K3,3 는 6개의 node의 complete bipartite graph.

4-color problem Guthrie, De Morgan 1852 G a connected planar graph with no bridges. Faces correspond to countries  a map

Every map is 4-colorable. Equivalent form: every planar graph is 4-colorable. Euler’s formula  every map contains a digon, triangle, square, or a pentagon. Kempe’s argument Final solution: Appel, Haken, (Univ. Illinois) using computer and Heesch’s discharging argument http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.html

Isomorphism problem When is two graph the same? Connectivity edge connectivity: the smallest number of edges whose removal disconnects G. Vertex connectivity: the above for vertices Various algorithms

Embedding problem (Planarity) 그래프를 어떤 곡면 선들이 만나지 않게 그려 넣을 수 있는가? 이때의 곡면은 무엇인가?