MTL 776: Graph Algorithms. B S Panda MZ 194

Transcription

1 MTL 776: Graph Algorithms B S Panda MZ 194 bspanda@maths.iitd.ac.in bspanda1@gmail.com

2 Lectre-1: Plan Definitions Types Terminology Sb-graphs Special types of Graphs Representations Graph Isomorphism

3 Definitions - Graph

4 Definitions Edge Type Directed: Ordered pair of vertices. Represented as (, v) directed from vertex to v. v Undirected: Unordered pair of vertices. Represented as {, v}. Disregards any sense of direction and treats both end vertices interchangeably. v

5 Definitions Edge Type Loop: A loop is an edge whose endpoints are eqal i.e., an edge joining a vertex to it self is called a loop. Represented as {, } = {} Mltiple Edges: Two or more edges joining the same pair of vertices.

6 Definitions Graph Type Simple (Undirected) Graph: consists of V, a nonempty set of vertices, and E, a set of nordered pairs of distinct elements of V called edges (ndirected) Representation Example: G(V, E), V = {, v, w}, E = {{, v}, {v, w}, {, w}} v w

7 Definitions Graph Type Mltigraph: G(V,E), consists of set of vertices V, set of Edges E and a fnction f from E to {{, v}, v V, v}. The edges e1 and e2 are called mltiple or parallel edges if f (e1) = f (e2). Representation Example: V = {, v, w}, E = {e 1,e 2,e 3 } e 1 e 2 w v e 3

8 Definitions Graph Type Psedograph: G(V,E), consists of set of vertices V, set of Edges E and a fnction f from E to {{, v}, v V}. Loops allowed in sch a graph. Representation Example: V = {, v, w}, E = {e 1,e 2,e 3,e 4 } e 1 e w e 4 2 v e 3

9 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 = {, v, w}, E = {(, v), (v, w), (w, )} v w

10 Definitions Graph Type Directed Mltigraph: G(V,E), consists of set of vertices V, set of Edges E and a fnction f from E to {(, v), v V}. The edges e1 and e2 are mltiple edges if f(e1) = f(e2) Representation Example: V = {, v, w}, E = {e 1,e 2,e 3,e 4 } e 1 e 2 e 4 e 3

11 Definitions Graph Type Type Edges Mltiple Edges Allowed? Loops Allowed? Simple Graph ndirected No No Mltigraph ndirected Yes No Psedograph ndirected Yes Yes Directed Graph directed No Yes Directed Mltigraph directed Yes Yes

12 Terminology Undirected graphs and v are adjacent if {, v} is an edge, e is called incident with and v. and v are called endpoints of {, v} Degree of Vertex (deg (v)): the nmber of edges incident on a vertex. A loop contribtes twice to the degree (why?). Pendant Vertex: deg (v) =1 Isolated Vertex: deg (v) = 0 Representation Example: For V = {, v, w}, E = { {, w}, {, w}, (, v) }, deg () = 2, deg (v) = 1, deg (w) = 1, deg (k) = 0, w and v are pendant, k is isolated v k w

13 Terminology Directed graphs For the edge (, v), is adjacent to vorvisadjacent from, Initial vertex, v Terminal vertex In-degree (deg - ()): nmber of edges for which is terminal vertex Ot-degree (deg + ()): nmber of edges for which is initial vertex Note: A loop contribtes 1 to both in-degree and ot-degree (why?) Representation Example: For V = {, v, w}, E = { (, w), ( v, w), (, v) }, deg - () = 0, deg + () = 2, deg - (v) = 1, deg + (v) = 1, and deg - (w) = 2, deg + () = 0 v w

14 Theorems: Undirected Graphs Theorem 1 The Handshaking theorem: 2e = d( v) v V (why?) Every edge connects 2 vertices

15 Theorems: Undirected Graphs Theorem 2: An ndirected graph has even nmber of vertices with odd degree Pr oof V1is the set of even degree vertices and V2 refers to odd degree vertices 2e = v V deg(v) = deg (v) is even for v V second term V v V deg() + v V Hence second term is also even deg(v) = even 2 1 the last ineqality is even since sm is 2e. 1, The sm of the last two terms on the right hand side of 2 deg(v) The first term in the right hand side of the last ineqality is even.

16 Theorems: directed Graphs Theorem 3: deg + () = deg - () = E

17 Simple graphs special cases Complete graph: K n, is the simple graph that contains exactly one edge between each pair of distinct vertices. Representation Example: K 1, K 2, K 3, K 4 K 1 K 2 K 3 K 4

18 Simple graphs special cases Cycle: C n, n 3 consists of n vertices v 1, v 2, v 3 v n and edges {v 1, v 2 }, {v 2, v 3 }, {v 3, v 4 } {v n-1, v n }, {v n, v 1 } Representation Example: C 3, C 4 C 3 C 4

19 Simple graphs special cases Wheels: W n, obtained by adding additional vertex to Cn and connecting all vertices to this new vertex by new edges. Representation Example: W 3, W 4 W 3 W 4

20 Simple graphs special cases N-cbes: Q n, 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: Q 1, Q Q 1 Q 2

21 Bipartite graphs In a simple graph G, if V can be partitioned into two disjoint sets V 1 and V 2 sch that every edge in the graph connects a vertex in V 1 and a vertex V 2 (so that no edge in G connects either two vertices in V 1 or two vertices in V 2 ) Application example: Representing Relations Representation example: V 1 ={v 1,v 2,v 3 }andv 2 ={v 4,v 5,v 6 }, v 1 v 4 v 2 v 5 v 3 v 6 V 1 V 2

22 Complete Bipartite graphs K m,n is the graph that has its vertex set portioned into two sbsets of m and n vertices, respectively There is an edge between two vertices if and only if one vertex is in the first sbset and the other vertex is in the second sbset. Representation example: K 2,3, K 3,3 K 2,3 K 3,3

23 Sbgraphs A sbgraph of a graph G = (V, E) is a graph H =(V, E ) where V is a sbset of V and E is a sbset of E Application example: solving sb-problems within a graph Representation example: V = {, v, w}, E = ({, v}, {v, w}, {w, }}, H 1, H 2 v w v w v G H 1 H 2

24 Sbgraphs G = G1 U G2 wherein E = E1 U E2 and V = V1 U V2, G, G1 and G2 are simple graphs of G Representation example: V1 = {, w}, E1 = {{, w}}, V2 = {w, v}, E1 = {{w, v}}, V = {, v,w}, E = {{{, w}, {{w, v}} w w v w v G1 G2 G

25 Representation Incidence (Matrix): Most sefl when information abot edges is more desirable than information abot vertices. Adjacency (Matrix/List): Most sefl when information abot the vertices is more desirable than information abot the edges. These two representations are also most poplar since information abot the vertices is often more desirable than edges in most applications

26 Representation- Incidence Matrix G=(V,E)beannditectedgraph.Spposethatv 1,v 2,v 3,,v n are the vertices and e 1,e 2,,e m are the edges of G. Then the incidence matrix with respect to this ordering of V and E is the nx mmatrixm=[m ij ], where m ij = 1 0 when edge e otherwise j is incident with v i Can also be sed to represent : Mltiple edges: by sing colmns with identical entries, since these edges are incident with the same pair of vertices Loops: by sing a colmn with exactly one entry eqal to 1, corresponding to the vertex that is incident with the loop

27 Representation- Incidence Matrix Representation Example: G = (V, E) e 1 e 2 e 3 e1 e2 v v e3 w w 0 1 1

28 Representation- Adjacency Matrix ThereisanNxNmatrix,where V =N,theAdjacenctMatrix (NxN) A = [a ij ] For ndirected graph a ij 1 = 0 if {v i, v j}is an edge of otherwise G For directed graph a ij 1 if (v i, v j) is an edge of = 0 otherwise G This makes it easier to find sbgraphs, and to reverse graphs if needed.

29 Representation- Adjacency Matrix Adjacency is chosen on the ordering of vertices. Hence, there as are as many as n! sch matrices. The adjacency matrix of simple graphs are symmetric (a ij = a ji ) (why?) When there are relatively few edges in the graph the adjacency matrix is a sparse matrix Directed Mltigraphs can be represented by sing aij = nmber of edges from v i to v j

30 Representation- Adjacency Matrix Example: Undirected Graph G (V, E) v w v v w w 1 1 0

31 Representation- Adjacency Matrix Example: directed Graph G (V, E) v w v v w w 1 0 0

32 Representation- Adjacency List Each node (vertex) has a list of which nodes (vertex) it is adjacent Example: ndirectd graph G (V, E) node Adjacency List v, w v w v w, w, v

33 Adjancency List Adjacency List

34 Definitions and Representation An ndirected graph and its adjacency matrix representation. An ndirected graph and its adjacency list representation.

35 Graph - Isomorphism G1 = (V1, E2) and G2 = (V2, E2) are isomorphic if: There is a one-to-one and onto fnction f from V1 to V2 with the property that a and b are adjacent in G1 if and only if f (a) and f (b) are adjacent in G2, for all a and b in V1. Fnction f is called isomorphism Application Example: In chemistry, to find if two componds have the same strctre

36 Graph - Isomorphism Representation example: G1 = (V1, E1), G2 = (V2, E2) f( 1 )=v 1,f( 2 )=v 4,f( 3 )=v 3,f( 4 )=v 2, 1 2 v 1 v v 3 v 4

37 Connectivity Basic Idea: In a Graph Reachability among vertices by traversing the edges Application Example: - In a city to city road-network, if one city can be reached from another city. - Problems if determining whether a message can be sent between two compter sing intermediate links - Efficiently planning rotes for data delivery in the Internet

38 Connectivity Path A Path is a seqence of edges that begins at a vertex of a graph and travels along edges of the graph, always connecting pairs of adjacent vertices. Representation example: G = (V, E), Path P represented, from to v is {{, 1}, {1, 4}, {4, 5}, {5, v}} v 4 5

39 Connectivity Path Definition for Directed Graphs A Path of length k (> 0) from to v in G is a seqence of k+1 vertices x1,x2, xk+1 sch that xixi+1 is an edge for i=1,2,,k-1. For Simple Graphs, seqence is x 0,x 1,,x n In directed mltigraphs when it is not necessary to distingish between their edges, we can se seqence of vertices to represent the path Circit/Cycle: = v, length of path > 0 Simple Path: does not contain a vertex more than once

40 Connectivity Connectedness Undirected Graph An ndirected graph is connected if there exists is a simple path between every pair of vertices Representation Example: G (V, E) is connected since for V = {v 1, v 2, v 3, v 4, v 5 }, there exists a path between {v i,v j }, 1 i, j 5 v 1 v 3 v 4 v 2 v 5

41 Connectivity Connectedness Undirected Graph Articlation Point (Ct vertex): removal of a vertex prodces a sbgraph with more connected components than in the original graph. The removal of a ct vertex from a connected graph prodces a graph that is not connected Ct Edge: An edge whose removal prodces a sbgraph with more connected components than in the original graph. Representation example: G (V, E), v 3 is the articlation point or edge {v 2,v 3 }, the nmber of connected components is 2 (> 1) v 1 v 3 v 5 v 2 v 4

42 Connectivity Connectedness Directed Graph A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph A directed graph is weakly connected if there is a (ndirected) path between every two vertices in the nderlying ndirected path A strongly connected Graph can be weakly connected bt the vice-versa is not tre (why?)

43 Connectivity Connectedness Directed Graph Representation example: G1 (Strong component), G2 (Weak Component), G3 is ndirected graph representation of G2 or G1 G1 G2 G3

44 Connectivity Connectedness Directed Graph Strongly connected Components: sbgraphs of a Graph G that are strongly connected Representation example: G1 is the strongly connected component in G G G1