LEC 1.pptx
- 1. Graph Theory Mr. Saleem Mustafa
- 2. Varying Applications (examples) • Computer networks • Distinguish between two chemical compounds with the same molecular formula but different structures • Solve shortest path problems between cities • Scheduling exams and assign channels to television stations
- 3. Topics Covered • Definitions • Types • Terminology • Representation • Sub-graphs • Connectivity • Hamilton and Euler definitions • Shortest Path • Planar Graphs • Graph Coloring
- 4. Definitions - Graph A generalization of the simple concept of a set of dots, links, edges or arcs. Representation: Graph G =(V, E) consists set of vertices denoted by V, or by V(G) and set of edges E, or E(G)
- 5. Definitions – Edge Type Directed: Ordered pair of vertices. Represented as (u, v) directed from vertex u to v. Undirected: Unordered pair of vertices. Represented as {u, v}. Disregards any sense of direction and treats both end vertices interchangeably. u v u v
- 6. Definitions – Edge Type • Loop: A loop is an edge whose endpoints are equal i.e., an edge joining a vertex to it self is called a loop. Represented as {u, u} = {u} • Multiple Edges: Two or more edges joining the same pair of vertices. u
- 7. Definitions – Graph Type Simple (Undirected) Graph: consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges (undirected) Representation Example: G(V, E), V = {u, v, w}, E = {{u, v}, {v, w}, {u, w}} u v w
- 8. Definitions – Graph Type Multigraph: G(V,E), consists of set of vertices V, set of Edges E and a function f from E to {{u, v}| u, v V, u ≠ v}. The edges e1 and e2 are called multiple or parallel edges if f (e1) = f (e2). Representation Example: V = {u, v, w}, E = {e1, e2, e3} u v w e1 e2 e3
- 9. Definitions – Graph Type Pseudograph: G(V,E), consists of set of vertices V, set of Edges E and a function F from E to {{u, v}| u, v Î V}. Loops allowed in such a graph. Representation Example: V = {u, v, w}, E = {e1, e2, e3, e4} u v w e1 e3 e2 e4
- 10. Definitions – Graph Type Directed Graph: G(V, E), set of vertices V, and set of Edges E, that are ordered pair of elements of V (directed edges) Representation Example: G(V, E), V = {u, v, w}, E = {(u, v), (v, w), (w, u)} u w v
- 11. Definitions – Graph Type Directed Multigraph: G(V,E), consists of set of vertices V, set of Edges E and a function f from E to {{u, v}| u, v V}. The edges e1 and e2 are multiple edges if f(e1) = f(e2) Representation Example: V = {u, v, w}, E = {e1, e2, e3, e4} u w v e1 e2 e3 e4
- 12. Definitions – Graph Type Type Edges Multiple Edges Allowed ? Loops Allowed ? Simple Graph undirected No No Multigraph undirected Yes No Pseudograph undirected Yes Yes Directed Graph directed No Yes Directed Multigraph directed Yes Yes
- 13. Terminology – Undirected graphs • u and v are adjacent if {u, v} is an edge, e is called incident with u and v. u and v are called endpoints of {u, v} • Degree of Vertex (deg (v)): the number of edges incident on a vertex. A loop contributes twice to the degree (why?). • Pendant Vertex: deg (v) =1 • Isolated Vertex: deg (k) = 0 Representation Example: For V = {u, v, w} , E = { {u, w}, {u, v} }, deg (u) = 2, deg (v) = 1, deg (w) = 1, deg (k) = 0, w and v are pendant , k is isolated u k w v
- 14. Terminology – Directed graphs • For the edge (u, v), u is adjacent to v OR v is adjacent from u, u – Initial vertex, v – Terminal vertex • In-degree (deg- (u)): number of edges for which u is terminal vertex • Out-degree (deg+ (u)): number of edges for which u is initial vertex Note: A loop contributes 1 to both in-degree and out-degree (why?) Representation Example: For V = {u, v, w} , E = { (u, w), ( v, w), (u, v) }, deg- (u) = 0, deg+ (u) = 2, deg- (v) = 1, deg+ (v) = 1, and deg- (w) = 2, deg+ (u) = 0 u w v
- 15. Theorems: Undirected Graphs Theorem 1 The Handshaking theorem: (why?) Every edge connects 2 vertices V v v e ) deg( 2
- 16. Theorems: Undirected Graphs Theorem 2: An undirected graph has even number of vertices with odd degree even V oof 2 2 1 V v 1, V v V u V v deg(u) term second even also is term second Hence 2e. is sum since even is inequality last the of side hand right on the terms last two the of sum The even. is inequality last the of side hand right in the first term The V for v even is (v) deg deg(v) deg(u) deg(v) 2e vertices degree odd to refers V2 and vertices degree even of set the is 1 Pr
- 17. Theorems: directed Graphs • Theorem 3: deg + (u) = deg - (u) = |E|
- 18. Simple graphs – special cases • Complete graph: Kn, is the simple graph that contains exactly one edge between each pair of distinct vertices. Representation Example: K1, K2, K3, K4 K2 K1 K4 K3
- 19. Simple graphs – special cases • Cycle: Cn, n ≥ 3 consists of n vertices v1, v2, v3 … vn and edges {v1, v2}, {v2, v3}, {v3, v4} … {vn-1, vn}, {vn, v1} Representation Example: C3, C4 C3 C4
- 20. Simple graphs – special cases • Wheels: Wn, obtained by adding additional vertex to Cn and connecting all vertices to this new vertex by new edges. Representation Example: W3, W4 W3 W4
- 21. Simple graphs – special cases • N-cubes: Qn, vertices represented by 2n bit strings of length n. Two vertices are adjacent if and only if the bit strings that they represent differ by exactly one bit positions Representation Example: Q1, Q2 0 10 1 00 11 Q1 01 Q2
- 22. Bipartite graphs • In a simple graph G, if V can be partitioned into two disjoint sets V1 and V2 such that every edge in the graph connects a vertex in V1 and a vertex V2 (so that no edge in G connects either two vertices in V1 or two vertices in V2) Application example: Representing Relations Representation example: V1 = {v1, v2, v3} and V2 = {v4, v5, v6}, v1 v2 v3 v4 v5 v6 V1 V2
- 23. Complete Bipartite graphs • Km,n is the graph that has its vertex set portioned into two subsets of m and n vertices, respectively There is an edge between two vertices if and only if one vertex is in the first subset and the other vertex is in the second subset. Representation example: K2,3, K3,3 K2,3 K3,3