1/24 Introduction to Graphs

2/24 Graph Definition Graph : consists of vertices and edges. Each edge must start and end at a vertex. Graph G = (V, E) –V = set of vertices –E = set of edges  (V  V) Example:

3/24 Graph Definition Degree of a vertex : number of edges connected to it Example: V 4 has degree = 3 V2V2 V5V5 V3V3 V4V4 V1V1

4/24 Graph Definition Total degree of a graph : sum of the degrees of all the vertices. Note: If a graph has n edges, the total degree = 2n. Example: graph has total degree = 10 V2V2 V3V3 V4V4 V1V1

5/24 Graph Definition The Handshaking Theorem: –Let G = (V, E). Then –Proof: Each edge contributes twice to the degree count of all vertices. –Example: If a graph has 5 vertices, can each vertex have degree 3? Or 4? –The sum is 5  3 = 15 which is an odd number. Not possible. –The sum is 5  4 = 20 = 2 |E| and 20/2 = 10 edges. May be possible.

6/24 Graph Definition Types of graphs –Undirected: edge (u, v) = (v, u); for all v, »(v, v)  E (usually no self loops) »Some may have self loops. –Directed: (u, v) is edge from u to v, denoted as u  v. Self loops are allowed. Is a graph where an edge represents a one-way relation only. –Cf. undirected graph – an edge represents two-way or symmetric relationship between two vertices. The number of directed edges which initiate from vertex v is called the outdegree of v or outdeg(v). The number of directed edges which terminate at vertex v is called the indegree of v or indeg(v).

7/24 Representation of Graphs Two standard ways –Adjacency Lists. –Adjacency Matrix. a dc b a b c d b a d dc c ab ac a dc b

8/24 Adjacency Lists Consists of an array Adj of |V| lists. One list per vertex. For u  V, Adj[u] consists of all vertices adjacent to u. a dc b a b c d b c d dc a dc b a b c d b a d dc c ab ac If weighted, also store weights in adjacency lists.

9/24 Storage Requirement For directed graphs: –Sum of lengths of all adj. lists is  out-degree(v) = |E| v  V –Total storage:  (|V|+|E|) For undirected graphs: –Sum of lengths of all adj. lists is  degree(v) = 2|E| v  V –Total storage:  (|V|+|E|) No. of edges leaving v No. of edges incident on v. Edge (u,v) is incident on vertices u and v.

10/24 Pros and Cons: adj list Pros –Space-efficient, when a graph is sparse. –Can be modified to support many graph variants. Cons –Determining if an edge (u,v)  G is not efficient. Have to search in u ’ s adjacency list.  (degree(u)) time.  (V) in the worst case.

11/24 Adjacency Matrix |V|  |V| matrix A. Number vertices from 1 to |V| in some arbitrary manner. A is then given by: a dc b a dc b A = A T for undirected graphs. If weighted, store weights also in adjacency matrix.

12/24 Space and Time Space:  (|V| 2 ). –Not memory efficient for large graphs. Time: to list all vertices adjacent to u:  (|V|). Time: to determine if (u, v)  E:  (1). Can store weights instead of bits for weighted graph.

13/24 Directed Graphs –Theorem –Examples (continued): Adjacency matrix of a directed graph: –outdeg(V 1 ) = 1, indeg(V 1 ) = 2 –outdeg(V 2 ) = 2, indeg(V 2 ) = 1 –outdeg(V 3 ) = 0, indeg(V 3 ) = 2 –outdeg(V 4 ) = 2, indeg(V 4 ) = 0

14/24 Each edge e has a weight, wt(e) graph may be undirected or directed weight may represent length, cost, capacity, etc adjacency matrix becomes weight matrix adjacency lists include weight in node v w y x z a weighted graph G Weighted Graphs

15/24 Vertex cover A set of vertices that cover all edges, i.e., at least one vertex of all edges is in the set. I.e., set of vertices W  V with for all {x,y}  E: x  W or y  W. Vertex Cover problem: –Given G, find vertex cover of minimum size Modified from

16/24 Subset S of vertices such that no two vertices in S are connected Independent Set (I)

17/24 Subset S of vertices such that no two vertices in S are connected Independent Set (II)

18/24 INSTANCE: graph G SOLUTION: independent set S in G MEASURE: maximize the size of S INSTANCE: graph G, number K QUESTION: does G have independent set of size  K OPTIMIZATION VERSION: DECISION VERSION: Independent Set (III)

19/24 Subset S of vertices such that every two vertices in S are connected Clique (I)

20/24 Clique INSTANCE: graph G, number K QUESTION: does G have a clique of size  K? Clique (II)

21/24 Relations The following are equivalent –G has an independent set with at least k vertices –The complement of G has a clique with at least k vertices –G has a vertex cover with at most n-k vertices

22/24 Bipartite Graphs A bipartite graph G is –a graph in which the vertices V can be partitioned into two disjoint subsets V 1 and V 2 such that no two vertices in V 1 are adjacent; no two vertices in V 2 are adjacent. A complete bipartite graph K m,n is –Is a bipartite graph in which the sets V 1 and V 2 contain m and n vertices, respectively, and every vertex in V 1 is adjacent to every vertex in V 2. –Q: How many edges ?

23/24 Assignment Problem Assignment problem. –Input: weighted, complete bipartite graph G = (L  R, E) with |L| = |R|. –Goal: find a perfect matching of min weight '2'3'4'5' Min cost perfect matching M = { 1-2', 2-3', 3-5', 4-1', 5-4' } cost(M) = = Introduction to Bioinformatics Algorithms

24/24 Assignment problem and related problems L: n jobs, R: m machines, weight of an edge (i,j) in E: time for job i taken on machine j. –Assign n jobs to each of m machines such that the total processing time is minimized. Matching: is a subset of E such that no two edges in the subset are adjacent. –Perfect matching: A matching saturates all vertices. –Maximum matching: A matching with the largest possible number of edges. Min-cost maximum matching problem: Find a max matching in a graph (with edge weights) such that the total edge weight is minimized.