Mohamed G. Gouda						     CS 313K
Summer 2013							     Midterm 2
One Hour

1) Let G be a graph with 7 vertices, named 1,...,7, and 14 edges, and let the
degrees of the vertices in G be as follows:
	deg(1) = 0
	deg(2) = 4
	deg(3) = 5
	deg(4) = 5
	deg(5) = 5
	deg(6) = 6
	deg(7) = 8
State 5 reasons for why the previous statement is false.

Sol.
Reason 1: G should satisfy Handshake Theorem (Twice the number of edges, 28, equals
the sum of degrees, 33).

Reason 2: The degree of any vertex should not be larger than (number of vertices - 1,
which is 6).

Reason 3: If the degree of any vertex, 6, is (number of vertices - 1), then the degree
of any other vertex, 1, should not be 0.

Reason 4: The sum of the degrees of all vertices, 33, should be even.

Reason 5: The number of vertices, whose degrees are odd, 3, should be even.

2) Let G=(V,E) be a connected graph that does not satisfy Euler's Formula (which implies
that G is nonplanar). Also let u be a vertex in V. Assume that a new vertex v and a new
edge (u,v) are added to G yielding a new graph G'=(V',E'). Prove that G' is nonplanar.

Sol.
Let R and R' be the sets of regions in graphs G and G', respectively. 

Because G does not satisfy Euler's Formula, we have (|V|-|E|+|R| != 2).

Also, |V'|=|V|+1, |E'| = |E|+1, |R'|=|R|. Therefore, 
	|V'|-|E'|+|R'| = |V|-|E|+|R| != 2.

Thus, G' does not satisfy Euler's Formula and is nonplanar.

3) Prove that the complete graph K5 is nonplanar.

Sol. 
For K5=(V,E), we have |V|=5, |E|=10, and for each region ri, rdeg(r)>=3 (by Region Degree
Theorem). 

From Euler's Formula, |R| = 2-|V|+|E| = 7

From Region Handshake Theorem, 20 = 2*|E| >= 3*|R| = 21, which is False.

Thus, K5 does not satisfy Euler's Formula or Region Handshake Theorem and is nonplananr.

4) An unconnected planar graph G consists of 2 connected components G1=(V1,E1) and 
G2=(V2,E2) where |V1|=5, |E1|=9, |V2|=6, |E2|=12. Compute the number of regions in G.

Sol. 
From Euler's Formula, we have |R1| = 2-|V1|+|E1| = 6 and |R2| = 2-|V2|+|E2| = 8.

Thus, |R| = |R1|+|R2|-1 = 6+8-1 = 13.