Combinatorial Computing: Week 1 - Graph Theory and Matching, Exams of Mathematics

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Mat 3770: Combinatorial Computing

Week 1

Spring 2009

Week 1

I (^) Syllabus and General Course Guidelines

I (^) Schedule (Tentative, but note exam dates)

I (^) Homework

I (^) Chapter 1. Elements of Graph Theory

Complete Graphs

I (^) If all possible edges are in E (i.e., every pair of vertices is connected), then G is called a complete graph.

I (^) A complete graph over n vertices is called Kn.

These are examples of the complete graphs K 3 and K 4.

Directed Graphs

I (^) If order of endpoints is important, then the edges of a graph are said to be directed edges.

I (^) A directed graph is one in which all edges are directed. (Also known as a digraph.)

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I (^) Vertex indegree : the number of directed edges coming in.

I (^) Vertex outdegree : the number of directed edges going out.

Connected Components

I (^) An undirected graph is connected if every pair of vertices is connected by a path.

fig a. fig b.

I (^) The connected components of a graph are the equivalence classes of vertices under the “is reachable from” relation. How many components are in each figure above?

I (^) A graph is connected if it has exactly one connected component.

Bipartite Graphs

I (^) A bipartite graph is an undirected graph G = fV ; E g in which V can be partitioned into two sets, V 1 and V 2 such that < u; w >∈ E implies either u ∈ V 1 and w ∈ V 2 , or u ∈ V 2 and w ∈ V 1.

I (^) Note: the result is that all edges have endpoints in both V 1 and V 2.

Maximum Matching

I (^) A maximum matching is a matching of maximum cardinality, that is, a matching M 3 ∀ matchings M ′; jM j  jM ′j. 1 2 3 4

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L R

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L R Bipartite Graph (^) a maximum a matching with matching cardinality 2

I (^) Suppose L is a set of machines with a set R of tasks to be performed simultaneously. We dene an edge < u; w > ∈ E to mean a particular machine u ∈ L is capable of performing a particular task w ∈ R.

Edge Cover

I (^) An edge cover is a set C of vertices in a graph G with the property that every edge of G is incident to at least one vertex in C (i.e., C contains at least one endpoint of every edge).

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I (^) An edge cover of an undirected graph G = (V ; E ) can also be dened as the subset V ′^ ⊆ V ; 3 if < u; w > ∈ E , then u ∈ V ′^ or w ∈ V ′, or both.

In other words, each vertex ìcoversî its incident edges, and an edge cover for G is a set of vertices that covers all the edges in E.

Independent Sets

I (^) An independent set of a graph G = (V ; E ) is a subset V ′^ ⊆ V 3 each edge in E is incident on at most one vertex in V ′^ (i.e., a set of vertices without an edge between any two of them).

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Maximal vs Maximum Independent Sets

I (^) A maximal independent set is a an independent set V ′^ 3 ∀ u ∈ V V ′, the set V ′^ ∪ fug is not independent (i.e., every vertex not in V ′^ is adjacent to some vertex in V ′. (An easy problem: can't add any more vertices and remain independent.)

I (^) A maximum independent set is an independent set of largest size in a general graph ó a very different and much harder problem!.

maximal:{g} maximum: {a, b, c, d, e, f}

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