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Mat 3770: Combinatorial Computing
Week 1
Spring 2009
1
Week 1
I (^) Syllabus and General Course Guidelines
I (^) Schedule (Tentative, but note exam dates)
I (^) Homework
I (^) Chapter 1. Elements of Graph Theory
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Graph Models
I (^) A graph , G = ( V ; E ), is a set of vertices or points ( V ) and the set of edges ( E ) connecting them.
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d c
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In this example, V = f a ; b ; c ; d ; e g, and E = f ab ; bc ; be ; ce g.
I (^) The degree of a vertex is the number of incident edges. What is the degree of each vertex in the figure above?
I (^) Two vertices are adjacent if there is an edge connecting them. Which pairs of vertices are adjacent in this figure?
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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.
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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.
Paths and Circuits
I A path P from vertex a to vertex b is a sequence of distinct vertices: P = f a = x 1 ; x 2 ; : : : ; xn = b g such that consecutive vertices are adjacent (i.e., < xi ; xi + 1 > ∈ E ).
I (^) The number of vertices in the path P is n ; the length of P (the number of edges in the path) is n 1.
I (^) A circuit is a path than ends where it starts (i.e., xn = x 1 ). (This single repetition is allowed, otherwise vertices are distinct.)
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.
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Bipartite Graphs
I (^) A bipartite graph is an undirected graph G = f V ; 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. 8
Matching
I (^) Given an undirected graph, G = ( V ; E ), a matching is a subset of edges M ⊆ E 3 ∀ vertices v ∈ V , at most one edge of M is incident on v (i.e., no two edges in M share an endpoint.)
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I (^) A vertex v ∈ V is matched by the matching M if some edge in M is incident on v ; otherwise, v is unmatched.
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Maximum Matching
I (^) A maximum matching is a matching of maximum cardinality, that is, a matching M 3 ∀ matchings M ′; j M j j M ′j. 1 2 3 4 5
a b c d L R
1 2 3 4 5
a b c d
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. 10
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.
Minimum Edge cover
I (^) The size of a cover is the number of vertices in it.
C 1 = {6, 3} C 2 = {1, 2, 4, 5}
(minimum)
