1. Part I: Introductory Materials Introduction to Graph Theory Dr. Nagiza F. Samatova Department of Computer Science North Carolina State University and Computer Science and Mathematics Division Oak Ridge National Laboratory

2. 2 Graphs Graph with 7 nodes and 16 edges Undirected Edges Nodes / VerticesDirected

3. 3 Types of Graphs Undirected vs. Directed Attributed/Labeled (e.g., vertex, edge) vs. Unlabeled Weighted vs. Unweighted General vs. Bipartite (Multipartite) Trees (no cycles) Hypergraphs Simple vs. w/ loops vs. w/ multi-edges

4. 4 Labeled Graphs and Induced Subgraphs Bold: A subgraph induced by vertices b, c and d Labeled graph w/ loops

5. Graph Isomorphism 5 Which graphs are isomorphic? (A)(B)(C) C

6. Graph Automorphism 6 Which graphs are automorphic? Automorphism is isomorphism that preserves the labels. (A)(B)(C) B

7. Vertex degree, in-degree, out-degree 77Directed head tail t h In-degree of the vertex is the number of in-coming edges Out-degree of the vertex is the number of out-going edges Degree of the vertex is the number of edges (both in- & out-degree)

8. 8 Graph Representation and Formats Adjacency Matrix (vertex vs. vertex) Incidence Matrix (vertex vs. edge) Sparse vs. Dense Matrices DIMACS file format In R: igraph object

9. 9 Adjacency Matrix Representation Representation is NOT unique. Algorithms can be order-sensitive. Src: “Introduction to Data Mining” by Kumar et al

10. Families of Graphs 10 Cliques Path and simple path Cycle Tree Connected graphs Read the book chapter for definitions and examples.

11. 11 Complete Graph, or Clique Each pair of vertices is connected.Clique

12. 12 The CLIQUE Problem Maximum Clique of Size 5 Clique Clique: a complete subgraph Maximal Clique Maximal Clique: a clique cannot be enlarged by adding any more vertices Maximum Clique Maximum Clique: the largest maximal clique in the graph

13. 13 Does this graph contain a 4-clique? Indeed it does! But, if it had not, what evidence would have been needed?

14. 14 Problem: Decision, Optimization or Search Problem Decision Optimization Search Formulate each version for the CLIQUE problem. (self-reduction) “Yes”-”No” Parameter k  max/min Actual solution Which problem is harder to solve? If we solve Decision problem, can we use it for the others? Enumeration All solutions

15. 15 Refresher: Class P and Class NP Definition: P (NP) is the class of languages/problems that are decidable in polynomial time on a (non-)deterministic single-tape Turing machine. Class P ???? NP non-polynomial Non- deterministic polynomial Polynomially verifiable

16. 16 PSPACE ∑ 2 P …… “forget about it” P vs. NP The Classic Complexity Theory View: P NP “easy” “hard” “About ten years ago some computer scientists came by and said they heard we have some really cool problems. They showed that the problems are NP-complete and went away!”

17. 17 Classical Graph Theory Problems CSC505:Algorithms, CSC707 :Complexity Theory, CSC5??:Graph Theory Longest Path Maximum Clique Minimum Vertex Cover Hamiltonian Path/Cycle Traveling Salesman (TSP) Maximum Independent Set Minimum Dominating Set Graph/Subgraph Isomorphism Maximum Common Subgraph … NP-hard Problems

18. 18 Graph Mining Problems CSC 422/522 and Our Book Clustering + Maximal Clique Enumeration Classification Association Rule Mining +Frequent Subgraph Mining Anomaly Detection Similarity/Dissimilarity/Distance Measures Graph-based Dimension Reduction Link Analysis … Many graph mining problems have to deal with classical graph problems as part of its data mining pipeline.

19. 19 Dealing with Computational Intractability Exact Algorithms: –Small graph problems –Small parameters to graph problems –Special classes of graphs (e.g., bounded tree-width) Approximation Polynomial-Time Algorithms (O(n c )) –Guaranteed error-bar on the solution Heuristic Polynomial-Time Algorithms –No guarantee on the quality of the solution –Low degree polynomial solutions Our focus