1. Graph & Trees Chapters 10-11
Discrete Mathematics
12/4/2018
Graph & Trees Chapters 10-11
Acknowledgement
This is a modified version of Module#22 on
Graph Theory by Michael Frank
Sultan Almuhammadi
ICS 254: Graphs and Trees
ICS 254: Graphs and Trees

2. What are Graphs?
Not
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.
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3. Applications of Graphs
Potentially 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, …
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4. §10.1: Definition of Graph A graph G = (V, E)
V is a nonempty set of verticies
E 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
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5. Visual Representation of a Simple Graph
Simple Graphs
A 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
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6. Example of a Simple Graph
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}}
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7. Multigraphs
Like 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
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8. Pseudographs
Like a multigraph, but edges connecting a node to itself are allowed. (R may be reflexive.)
A pseudograph G=(V, E) where Edge eE is a loop if
e = {u, u} = {u}.
E.g., nodes are the cities in a state, edges are the roads serving/connecting the cities.
loops
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9. Directed Graphs
Correspond 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}
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10. Directed Multigraphs
Like 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...
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11. Types of Graphs: Summary
Summary of the book’s definitions.
Keep in mind this terminology is not fully standardized across different authors...
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12. §10.2: Graph Terminology You need to learn the following terms:
Adjacent, connects, endpoints, degree, initial, terminal, in-degree, out-degree, complete, cycles, wheels, n-cubes, bipartite, subgraph, union.
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13. Adjacency
Let 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.
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14. Degree of a Vertex Let 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.
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15. Exercise Find: An element of degree 4 An element of degree 3 A pendant
An isolated vertix
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16. Handshaking Theorem
Let 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.
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17. Directed Adjacency
Let 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 u
e comes from u, e goes to v.
e connects u to v, e goes from u to v
the initial vertex of e is u
the terminal vertex of e is v
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18. Directed Degree Let 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.
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19. Directed Handshaking Theorem
Let 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.
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20. Special Graph Structures
Special cases of undirected graph structures:
Complete graphs Kn
Cycles Cn
Wheels Wn
n-Cubes Qn
Bipartite graphs
Complete bipartite graphs Km,n
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21. Complete Graphs
For any nN, a complete graph on n vertices, Kn, 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. K1 K4 K2 K3 K5 K6
Note that Kn has edges.
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22. How many edges are there in Cn?
Cycles
For any n3, a cycle on n vertices, Cn, is a simple graph where V={v1,v2,… ,vn} and E={{v1,v2},{v2,v3},…,{vn1,vn},{vn,v1}}. C3 C4 C5 C6 C8 C7
How many edges are there in Cn?
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23. How many edges are there in Wn?
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},…,{vhub,vn}}. W3 W4 W5 W6 W8 W7
How many edges are there in Wn?
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24. Number of vertices: 2n. Number of edges:Exercise to try!
n-cubes (hypercubes)
For any nN, the hypercube Qn is a simple graph consisting of two copies of Qn-1 connected together at corresponding nodes. Q0 has 1 node. Q0 Q1 Q2 Q4 Q3
Number of vertices: 2n. Number of edges:Exercise to try!
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25. Bipartite Graphs
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 parts in such a way that all edges go between the two parts. V1 V2
This definition can easily be adapted for the case of multigraphs and directed graphs as well.
Can represent with zero-one matrices.
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26. Bipartite Graphs Exer: Are these graphs bipartite? G H
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27. Complete Bipartite Graphs
For m,nN, the complete bipartite graph Km,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.
K4,3
Km,n has _____ nodes and _____ edges.
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28. Subgraphs
A subgraph of a graph G=(V,E) is a graph H=(W,F) where WV and FE. G H
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29. Graph Unions
The union G1G2 of two simple graphs G1=(V1, E1) and G2=(V2,E2) is the simple graph (V1V2, E1E2). a b c d e  a b c d f
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30. §10.3: Graph Representations & Isomorphism
Adjacency lists.
Adjacency matrices.
Incidence matrices.
Graph isomorphism:
Two graphs are isomorphic iff they are identical except for their node names.
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31. Adjacency Lists
A table with 1 row per vertex, listing its adjacent vertices. b a d c e f
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32. Directed Adjacency Lists
1 row per node, listing the terminal nodes of each edge incident from that node.
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33. Adjacency Matrices A way to represent simple graphs
possibly with self-loops.
Matrix A=[aij], where aij 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.
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34. Graph Isomorphism Formal definition:
Simple graphs G1=(V1, E1) 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.
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35. Graph Invariants under Isomorphism
Necessary but not sufficient conditions for G1=(V1, E1) 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.
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36. Isomorphism Example
If isomorphic, label the 2nd graph to show the isomorphism, else identify difference. d b a b a d c e e c f f
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37. Are These Isomorphic?
If isomorphic, label the 2nd graph to show the isomorphism, else show an invariant property.
Same # of vertices a b
Same # of edges
Different # of vertices of degree 2 (1 vs 3) d e c
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38. §10.4: Connectivity
A 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.
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39. The length of the path the length of the path p
|p| = number of edges in p
e.g.
p = (a,b), (b,c), (c,d)
|p| = 3
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40. Connected Graphs
An undirected graph is connected iff there is a path between every pair of distinct vertices in the graph.
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41. Directed Connectedness
A 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.
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42. §10.5: Euler & Hamilton Paths
E.g. Can we walk through the town below, crossing each bridge exactly once, and return to start? A D B C
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43. §10.5: Euler & Hamilton Paths
Euler circuit: a simple circuit containing every edge (exactly once)
Euler path: in G is a simple path containing every edge (exactly once)
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44. The Euler Circuit/Path
Can we walk through the town below, crossing each bridge exactly once? (and return to start?) A D B C
Dual multigraph
The original problem
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45. The Euler Circuit/Path
Exer:
Does K4 have an EC? EP?
Does K5 have an EC? EP?
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46. Euler Path Theorems
Thrm 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.
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47. §10.5: Euler & Hamilton Paths
A Hamilton circuit is a circuit that traverses each vertex 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?
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48. §10.7: Planar Graphs
Defn. G is plannar if it can be drawn on the plane without crossing edges.
Eg. K3, K4, Q2
Exer: Is K3,3 plannar?
Exer: Is K5 plannar?
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49. §10.8: Graph Coloring
Coloring 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:
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50. §10.8: Graph Coloring
The chromatic number of G, denoted by χ(G), is the minimum number of colors G can have.
E.g. Find χ(G). G:
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51. Graph Coloring Example
E.g. Find χ(W5)
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52. Exer: Graph Coloring Exer: Find χ(G) where G is: W8 K5 K5,3
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53. Graph Coloring Theorems on χ(G): If G is bipartite, then χ(G) ≤ 2
If G is a plannar graph, then
χ(G) ≤ 4 (Map coloring of the dual graph*)
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54. §11.1: Introduction to Trees
A 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
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55. Tree and Forest Examples
Leaves in green, internal nodes in brown.
A Tree:
A Forest:
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56. Rooted Trees
A 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.
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57. Same tree except for choice of root
Rooted Tree Examples
Note that a given unrooted tree with n nodes yields n different rooted trees.
Same tree except for choice of root
root
root
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58. Rooted-Tree Terminology Exercise
Find the parent, children, siblings, ancestors, & descendants of node f. o n h r d m b
root a c g e q i f l p j k
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59. n-ary trees
A rooted tree is called n-ary if every vertex has no more than n children.
e.g. binary tree for n = 2
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60. Rooted-Tree Terminology
Definition: 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.
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61. §11.3: Tree Traversal Preorder traversal Inorder traversal
Root, then subtrees (left-to-right)
Inorder traversal
Left subtree, root, then other subtrees
Postorder traversal
Subtrees (left-to-right) before the root.
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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^x2
Postfix form (reverse polish notation):
xy+x2^/
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