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Introduction to Graph Theory

Introduction to Graph Theory. Lecture 19: Digraphs and Networks. Introduction. Diagraph : A directed graph with directions on the edges. Digraph are used to model problems where the direction of flow of some quantity is important.

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Introduction to Graph Theory

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  1. Introduction to Graph Theory Lecture 19: Digraphs and Networks

  2. Introduction • Diagraph: A directed graph with directions on the edges. • Digraph are used to model problems where the direction of flow of some quantity is important. • Network: a digraph with limits placed on the quantity flown through a particular directed edge.

  3. Directed Graph • A digraph consists of a finite nonempty set of vertices V(D) and a set of ordered pairs of distinct vertices called arcs. • For the graph below, • the arc xy goes from x to y. • x is adjacent toy • y is adjacent from x x y

  4. More Terminology • od(v): outdegree. The number of vertices that v is adjacent to. • id(v): indegree. The number of vertices that v is adjacent from. • Transmitter: a vertex v with id(v)=0. (only sending information, but receiving none) • Receiver: a vertex v with in(v)=0.

  5. Our first simple theorem • Theorem 10.1: If D is a digraph with vertex set and having q arcs, then

  6. Connectivity • Weakly connected: A connected digraph with pairs of vertices not accessible from each other. • Unilateral: for every pair of vertices (u,v) there is either a directed u-v or v-u path. • Strongly connected: for every pair of vertices (u,v) there are both directed u-v and v-u paths.

  7. Strong Orientation • Assigning a direction to each edge is orienting the graph. • If the resulting graph becomes strongly connected, it is strong orientation. • Q1: Is there any way that we could make a graph with a bridge strongly oriented? • Q2: Is strong orientation always possible if G contains no bridge?

  8. Achieving Strong Orientation • Theorem 10.2: A connected graph G has a strong orientation if and only if it contains no bridges, i.e. every edge is in some cycle. • To obtain a strong orientation, we use our favorite search strategy --- DFS • Obtain a tree T using DFS • Orient each edge of T toward the vertex with higher number • Orient the remaining edges of G toward the vertex with lower number.

  9. Example • Can we convince that every vertex can reach the root?

  10. Acyclic Dgraphs • A digraph that has no directed cycle is called acyclic. • Theorem 10.3: Every acyclic digraph has at least one vertex of outdegree zero and at least one vertex of indegree zero. • Proof: • Consider the last vertex v in any longest path in the digraph, od(v)=0 • Consider the first vertex u of a longest path P, id(v)=0 u w x y v

  11. Applications of Acyclic Digraphs • A partially ordered set (poset) is often modeled by using an acyclic diagraph. • A partial order on a set is a relation that is • Reflexive (a~a, “a is related to a”) • Antisymmetric (a~b and b~a implies a=b) • Transitive (a~b and b~c implies a~c) • Examples of such relations are “less equal” and “a subset of”

  12. Example • Consider the set A={2,3,5,6,10,12,15,39} with the relation | (divide) • Is the relation a partial order • If so, draw the associated acyclic digraph • What is the acyclic digraph if we omit the transitive arcs? • Do you verify theorem 103 with this poset?

  13. Tournaments • Since some of you are volleyball player, this topic might interest you. • There are two kinds of tournament: • Elimination tournament --- once a team loses a game, it is out of the competition • Round-robin tournament --- each team plays each other team exactly once. • We’ll focus out discussion on round-robin tournaments.

  14. Round-Robin Tournament • A tournament is a directed graph. • A tournament on n vertices is an orientation of Kn. • An arc from u to v indicates that vertex u defeated vertex v. • How many possible outcome for a tournament of 3 teams and 4 teams? • What we would like to do is to rank the players from best to worst, which is a hard task • Consider a tournament of 5 teams

  15. (cont) • However, it is still possible to arrange the players on a list so that player i beats player i+1 for . • The next theorem should convince us the statement. • Theorem 10.4: Every Tournament contains a directed hamiltonian path.

  16. Proof of Theorem 10.4 • Proof by induction • Basic case: True for • Hypothesis: True for every tournament with n=k • We want to prove that it is true for a tournament with k+1 teams. • Let’s consider the the T-v for any v. there is a h-path . • Let vi be the first vertex for which , then the h-path in T is • If no such vi then the h-path is

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