Chapters 10.1 and 10.2 Based on slides by Y. Peng University of Maryland. Graphs. Introduction to Graphs. Definition: A simple graph G = (V, E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges.
Solution: Vertex f is isolated, and vertices a, d and j are pendant. The maximum degree is deg(g) = 5. This graph is a pseudograph (undirected, loops).
Result: There are 9 edges, and the sum of all degrees is 18. This is easy to explain: Each new edge increases the sum of degrees by exactly two.
deg-(a) = 1
deg+(a) = 2
deg-(b) = 4
deg+(b) = 2
deg-(d) = 2
deg+(d) = 1
deg-(c) = 0
deg+(c) = 2
No, because there is no way to partition the vertices into two sets so that there are no edges with both endpoints in the same set.
Example II: Is C6 bipartite?
Yes, because we can display C6 like this:
subgraph of K5
G1 G2 = K5
A matching M in a simple graph G = (V ,E) is a subset of the set E of edges of the graph such that no two edges are incident with the same vertex. A vertex that is the endpoint of an edge of a matching M is said to be matched in M; otherwise it is said to be unmatched.
A maximum matching is a matching with the largest number of edges. A matching M in a bipartite graph G = (V ,E) with bipartition (V1, V2) is a complete matching from V1 to V2 if every vertex in V1 is the endpoint of an edge in the matching, or equivalently, if |M| = |V1|.
For example, to assign jobs to employees so that the largest number of jobs are assigned employees, we seek a maximum matching. To assign employees to all jobs we seek a complete matching from the set of jobs to the set of employees.
HALL’S MARRIAGE THEOREM (1935)
The bipartite graph G = (V ,E) with bipartition (V1, V2) has a complete matching from V1 to V2
if and only if
|N(A)| ≥ |A| for all subsets A of V1.