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Introduction to Digraphs and Networks in Graph Theory

Learn about directed graphs and networks for modeling flow problems. Explore connectivity, orientation, tournaments, and acyclic digraphs with applications. Understand theorems and their proofs in graph theory.

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Introduction to Digraphs and Networks in 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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