Sultan Almuhammadi ICS 254: Graphs and Trees 1 Graph & Trees - PowerPoint Presentation

Sultan Almuhammadi ICS 254: Graphs and Trees 1 Graph & TreesChapters 10-11 Acknowledgement This is a modified version of Module#22 on Graph Theory by Michael Frank

Sultan Almuhammadi ICS 254: Graphs and Trees 2 What are Graphs?General meaning in everyday math: A plot or chart of numerical data using a coordinate system.Technical meaning in discrete mathematics:A particular class of discrete structures (to be defined) that is useful for representing relations and has a convenient webby-looking graphical representation. Not

Sultan Almuhammadi ICS 254: Graphs and Trees 3 Applications of GraphsPotentially anything (graphs can represent relations, relations can describe the extension of any predicate).Apps in networking, scheduling, flow optimization, circuit design, path planning.More apps: Geneology analysis, computer game-playing, program compilation, object-oriented design, …

Sultan Almuhammadi ICS 254: Graphs and Trees 4 §10.1: Definition of GraphA graph G = (V, E)V is a nonempty set of verticiesE is a set of edges (arcs or links) between vertices Type of edges: Directed: ordered pairs (u, v) of vertices Undirected: unordered pairs {u, v} of vertices

Sultan Almuhammadi ICS 254: Graphs and Trees 5 Simple GraphsA simple graph G =( V , E ) consists of:a set V of vertices or nodes (V corresponds to the universe of the relation R),a set E of edges: unordered pairs of [distinct] vertices u,v  V Visual Representation of a Simple Graph

Sultan Almuhammadi ICS 254: Graphs and Trees 6 Let V be the set of the 6 GCC states:V={KSA, UAE, KWT, OMN, QTR, BAH}Let E ={{ u , v }| u adjoins v}E ={{KSA,UAE}, {KSA,KWT}, {KSA,BAH}, {KSA,QTR}, {UAE,OMN}}Example of a Simple Graph

Sultan Almuhammadi ICS 254: Graphs and Trees 7 MultigraphsLike simple graphs, but there may be more than one edge connecting two given nodes.E.g. , nodes are cities, edges are segments of major highways between the cities. Parallel edges

Sultan Almuhammadi ICS 254: Graphs and Trees 8 PseudographsLike a multigraph, but edges connecting a node to itself are allowed. (R may be reflexive.) A pseudograph G =( V , E) whereEdge eE is a loop if e = {u, u} = {u} .E.g., nodes are the cities in a state, edges arethe roads serving/connectingthe cities. loops

Sultan Almuhammadi ICS 254: Graphs and Trees 9 Directed GraphsCorrespond to arbitrary binary relations R, which may not be symmetric.A directed graph (V , E ) consists of a set of vertices V and a binary relation E on V.E.g.: V = set of People,E={(x,y) | x loves y}

Sultan Almuhammadi ICS 254: Graphs and Trees 10 Directed MultigraphsLike directed graphs, but there may be more than one arc from a node to another.E.g., V=web pages, E =hyperlinks. The WWW is a directed multigraph...

Sultan Almuhammadi ICS 254: Graphs and Trees 11 Types of Graphs: SummarySummary of the book’s definitions.Keep in mind this terminology is not fully standardized across different authors...

Sultan Almuhammadi ICS 254: Graphs and Trees 12 §10.2: Graph TerminologyYou need to learn the following terms:Adjacent, connects, endpoints, degree, initial, terminal, in-degree, out-degree, complete, cycles, wheels, n-cubes, bipartite, subgraph, union.

Sultan Almuhammadi ICS 254: Graphs and Trees 13 AdjacencyLet G be an undirected graph with edge set E. Let e  E be (or map to) the pair { u , v}. Then we say:u, v are adjacent / neighbors / connected.Edge e is incident with vertices u and v .Edge e connects u and v.Vertices u and v are endpoints of edge e.

Sultan Almuhammadi ICS 254: Graphs and Trees 14 Degree of a VertexLet G be an undirected graph, v V a vertex. The degree of v , deg(v), is its number of incident edges. (Except that any self-loops are counted twice.)A vertex with degree 0 is called isolated.A vertex of degree 1 is called pendant.

Sultan Almuhammadi ICS 254: Graphs and Trees 15 ExerciseFind:An element of degree 4An element of degree 3A pendantAn isolated vertix

Sultan Almuhammadi ICS 254: Graphs and Trees 16 Handshaking TheoremLet G be an undirected (simple, multi-, or pseudo-) graph with vertex set V and edge set E. Then Corollary: Any undirected graph has an even number of vertices of odd degree.

Sultan Almuhammadi ICS 254: Graphs and Trees 17 Directed AdjacencyLet G be a directed (possibly multi-) graph, and let e be an edge of G that is (or maps to) (u , v ). Then we say: u is adjacent to v, v is adjacent from ue comes from u, e goes to v.e connects u to v, e goes from u to vthe initial vertex of e is uthe terminal vertex of e is v

Sultan Almuhammadi ICS 254: Graphs and Trees 18 Directed DegreeLet G be a directed graph, v a vertex of G.The in-degree of v , deg  (v), is the number of edges going to v.The out-degree of v, deg(v), is the number of edges coming from v.The degree of v, deg( v):deg(v)+deg(v), is the sum of v’s in-degree and out-degree.

Sultan Almuhammadi ICS 254: Graphs and Trees 19 Directed Handshaking TheoremLet G be a directed (possibly multi-) graph with vertex set V and edge set E. Then: Note that the degree of a node is unchanged by whether we consider its edges to be directed or undirected.

Sultan Almuhammadi ICS 254: Graphs and Trees 20 Special Graph StructuresSpecial cases of undirected graph structures:Complete graphs Kn Cycles C n Wheels W n n-Cubes QnBipartite graphsComplete bipartite graphs Km,n

Sultan Almuhammadi ICS 254: Graphs and Trees 21 Complete GraphsFor any nN , a complete graph on n vertices, K n, is a simple graph with n nodes in which every node is adjacent to every other node: u,vV: uv{u,v}E. K 1 K 2 K 3 K 4 K 5 K 6 Note that K n has edges.

Sultan Almuhammadi ICS 254: Graphs and Trees 22 CyclesFor any n3, a cycle on n vertices, C n , is a simple graph where V ={v1,v2,… ,vn} and E={{v1,v2},{v2,v3},…,{vn 1,vn},{vn,v1}}. C 3 C 4 C 5 C 6 C 7 C 8 How many edges are there in C n ?

Sultan Almuhammadi ICS 254: Graphs and Trees 23 Wheels For any n 3 , a wheel Wn, is a simple graph obtained by taking the cycle Cn and adding one extra vertex vhub and n extra edges {{vhub,v1}, {vhub,v2},…,{v hub,vn}}. W 3 W 4 W 5 W 6 W 7 W 8 How many edges are there in W n ?

Sultan Almuhammadi ICS 254: Graphs and Trees 24 n-cubes (hypercubes)For any n N , the hypercube Q n is a simple graph consisting of two copies of Q n -1 connected together at corresponding nodes. Q0 has 1 node. Q 0 Q 1 Q 2 Q 3 Q 4 Number of vertices: 2 n . Number of edges:Exercise to try!

Sultan Almuhammadi ICS 254: Graphs and Trees 25 Def’n.: A graph G=(V,E ) is bipartite (two-part) iff V = V1  V2 where V1 ∩ V2= and eE: v1V1,v2V2 : e={v1,v2}.In English: The graph can be divided into two partsin such a way that all edges go between the two parts. Bipartite Graphs V 1 V 2 This definition can easily be adapted for the case of multigraphs and directed graphs as well. Can represent with zero-one matrices.

Sultan Almuhammadi ICS 254: Graphs and Trees 26 Exer: Are these graphs bipartite?Bipartite Graphs G H

Sultan Almuhammadi ICS 254: Graphs and Trees 27 Complete Bipartite GraphsFor m,n  N , the complete bipartite graph K m,n is a bipartite graph where |V1| = m, |V2| = n, and E = {{v1,v2}|v1V1  v2V2}.That is, there are m nodes in the left part, n nodes in the right part, and every node in the left part is connected to every node in the right part. K 4,3 K m , n has _____ nodes and _____ edges.

Sultan Almuhammadi ICS 254: Graphs and Trees 28 SubgraphsA subgraph of a graph G=(V , E ) is a graph H =( W,F) where WV and FE. G H

Sultan Almuhammadi ICS 254: Graphs and Trees 29 Graph UnionsThe union G1  G 2 of two simple graphs G 1 =(V1, E1) and G2=(V2,E2) is the simple graph (V1V2, E 1E2). a b c d e a b c d f 

Sultan Almuhammadi ICS 254: Graphs and Trees 30 §10.3: Graph Representations & IsomorphismGraph representations:Adjacency lists.Adjacency matrices.Incidence matrices. Graph isomorphism: Two graphs are isomorphic iff they are identical except for their node names.

Sultan Almuhammadi ICS 254: Graphs and Trees 31 Adjacency ListsA table with 1 row per vertex, listing its adjacent vertices. a b d c f e

Sultan Almuhammadi ICS 254: Graphs and Trees 32 Directed Adjacency Lists1 row per node, listing the terminal nodes of each edge incident from that node.

Sultan Almuhammadi ICS 254: Graphs and Trees 33 Adjacency MatricesA way to represent simple graphspossibly with self-loops.Matrix A=[ a ij ] , where a ij is 1 if {vi, vj} is an edge of G, and is 0 otherwise.Can extend to pseudographs by letting each matrix elements be the number of links (possibly >1) between the nodes.

Sultan Almuhammadi ICS 254: Graphs and Trees 34 Graph IsomorphismFormal definition:Simple graphs G1 =( V 1 , E 1 ) and G2=(V2, E2) are isomorphic iff  a bijection f:V1V2 such that  a,bV1, a and b are adjacent in G1 iff f(a) and f(b) are adjacent in G2. f is the “renaming” function between the two node sets that makes the two graphs identical.This definition can easily be extended to other types of graphs.

Sultan Almuhammadi ICS 254: Graphs and Trees 35 Graph Invariants under IsomorphismNecessary but not sufficient conditions for G 1 =( V 1 , E 1) to be isomorphic to G2=(V2, E2):We must have that |V1|=|V2|, and |E1|=|E2|.The number of vertices with degree n is the same in both graphs.For every proper subgraph g of one graph, there is a proper subgraph of the other graph that is isomorphic to g.

Sultan Almuhammadi ICS 254: Graphs and Trees 36 Isomorphism ExampleIf isomorphic, label the 2nd graph to show the isomorphism, else identify difference. a b c d e f b d a e f c

Sultan Almuhammadi ICS 254: Graphs and Trees 37 Are These Isomorphic?If isomorphic, label the 2nd graph to show the isomorphism, else show an invariant property. a b c d e Same # of vertices Same # of edges Different # of vertices of degree 2 (1 vs 3)

Sultan Almuhammadi ICS 254: Graphs and Trees 38 §10.4: ConnectivityA path, p, from u to v is a sequence of adjacent edges going from vertex u to vertex v . e.g. p = ( u,x ), ( x,y ), …, (z,v)A path is a circuit if u=v.A path traverses the vertices along it.A path is simple if it contains no edge more than once.

Sultan Almuhammadi ICS 254: Graphs and Trees 39 The length of the paththe length of the path p |p| = number of edges in pe.g. p = (a,b), (b,c), (c,d) |p| = 3

Sultan Almuhammadi ICS 254: Graphs and Trees 40 Connected GraphsAn undirected graph is connected iff there is a path between every pair of distinct vertices in the graph.

Sultan Almuhammadi ICS 254: Graphs and Trees 41 Directed ConnectednessA directed graph is strongly connected iff there is a directed path from a to b for any two vertices a and b. It is weakly connected iff the underlying undirected graph (i.e., with edge directions removed) is connected.Note strongly implies weakly but not vice-versa.

Sultan Almuhammadi ICS 254: Graphs and Trees 42 §10.5: Euler & Hamilton PathsE.g. Can we walk through the town below, crossing each bridge exactly once, and return to start? A B C D

12/13/2015 ICS 254: Graphs and Trees 43 §10.5: Euler & Hamilton Paths E uler circuit: a simple circuit containing every e dge (exactly once) E uler path: in G is a simple path containing every edge (exactly once)Sultan Almuhammadi ICS 254: Graphs and Trees43

12/13/2015 ICS 254: Graphs and Trees 44 The Euler Circuit/Path Can we walk through the town below, crossing each bridge exactly once? (and return to start?) A B C D The original problem Dual multigraph Sultan Almuhammadi ICS 254: Graphs and Trees 44

12/13/2015 ICS 254: Graphs and Trees 45 The Euler Circuit/Path Exer : Does K 4 have an EC? EP? Does K 5 have an EC? EP? Sultan AlmuhammadiICS 254: Graphs and Trees45

Sultan Almuhammadi ICS 254: Graphs and Trees 46 Euler Path TheoremsThrm 1: A connected multigraph has an Euler circuit iff each vertex has even degree. Thrm 2: A connected multigraph has an Euler path (but not an Euler circuit) iff it has exactly 2 vertices of odd degree. One is the start, the other is the end.

12/13/2015 ICS 254: Graphs and Trees 47 §10.5: Euler & Hamilton Paths A Hamil t on circuit is a circuit that traverses each ver t ex in G exactly once.A Hamilton path is a path that traverses each vertex in G exactly once. E.g. Does W5 have an HC? HP?Sultan AlmuhammadiICS 254: Graphs and Trees47

Sultan Almuhammadi ICS 254: Graphs and Trees 48 §10.7: Planar GraphsDefn. G is plannar if it can be drawn on the plane without crossing edges. Eg . K 3 , K 4 , Q 2Exer: Is K3,3 plannar?Exer: Is K5 plannar?

Sultan Almuhammadi ICS 254: Graphs and Trees 49 §10.8: Graph ColoringColoring of a simple graph G=(V,E) is a mapping f:V A, where A = { 1 , 2 ,.., n } is a set of colors, such that for every edge (x,y) in E, f(x) ≠ f(y) i.e. Adjacent vertices have different colors G:

Sultan Almuhammadi ICS 254: Graphs and Trees 50 §10.8: Graph ColoringThe chromatic number of G, denoted by χ (G), is the minimum number of colors G can have . E.g. Find χ(G). G:

12/13/2015 ICS 254: Graphs and Trees 51 Graph Coloring Example E.g. Find χ ( W 5 )Sultan Almuhammadi ICS 254: Graphs and Trees51

12/13/2015 ICS 254: Graphs and Trees 52 Exer: Graph Coloring Exer : Find χ (G) where G is: W 8K5K 5,3Sultan AlmuhammadiICS 254: Graphs and Trees52

12/13/2015 ICS 254: Graphs and Trees 53 Graph Coloring Theorems on χ (G ) : If G is bipartite, then χ(G) ≤ 2If G is a plannar graph, then χ(G) ≤ 4 (Map coloring of the dual graph*)Sultan AlmuhammadiICS 254: Graphs and Trees53

Sultan Almuhammadi ICS 254: Graphs and Trees 54 §11.1: Introduction to TreesA tree is a connected undirected graph that contains no circuits. Thrm1: There is a unique simple path between any two of its nodes. Thrm2: T=(V,E) is a tree iff |E| = |V| - 1 A (not-necessarily-connected) undirected graph without simple circuits is called a forest.A forest is a bunch of trees

Sultan Almuhammadi ICS 254: Graphs and Trees 55 Tree and Forest ExamplesA Tree:A Forest: Leaves in green, internal nodes in brown.

Sultan Almuhammadi ICS 254: Graphs and Trees 56 Rooted TreesA rooted tree is a tree in which one node has been designated the root.Every edge is directed away from the root. Keywords about rooted trees: Parent, child, siblings, ancestors, descendents , leaf, internal node, subtree.

Sultan Almuhammadi ICS 254: Graphs and Trees 57 Rooted Tree ExamplesNote that a given unrooted tree with n nodes yields n different rooted trees. root root Same tree except for choice of root

Sultan Almuhammadi ICS 254: Graphs and Trees 58 Rooted-Tree Terminology ExerciseFind the parent,children, siblings,ancestors, & descendants of node f. a b c d e f g h i j k l m n o p q r root

Sultan Almuhammadi ICS 254: Graphs and Trees 59 n-ary treesA rooted tree is called n-ary if every vertex has no more than n children. e.g. binary tree for n = 2

Sultan Almuhammadi ICS 254: Graphs and Trees 60 Rooted-Tree TerminologyDefinition: The level of a node is the length of the simple path from the root to the node.The height of a tree is maximum node level.

Sultan Almuhammadi ICS 254: Graphs and Trees 61 §11.3: Tree TraversalPreorder traversalRoot, then subtrees (left-to-right)Inorder traversal Left subtree, root, then other subtrees Postorder traversal Subtrees (left-to-right) before the root.

12/13/2015 ICS 254: Graphs and Trees 62 Infix/prefix/postfix notation E.g. Draw the rooted tree for: (x + y) / (x^2) Infix form: ( x+y ) / (x^2) (need parentheses)Prefix form (polish notation): /+xy^x2Postfix form (reverse polish notation): xy+x2^/Sultan AlmuhammadiICS 254: Graphs and Trees62