Question

a.

An edge in an undirected connected graph is a bridge if removing it disconnects the graph. Prove that every connected graph all of whose vertices have even degrees contains no bridges.

b.Let r,s,u be binary relations in U. Verify the following property: if both relations r and s are transitive then the intersection of r and s is transitive too.

Answer #1

Suppose that H is a connected graph that
contains a proper cycle. Argue that removing any single
edge from this cycle will leave a subgraph of H
that remains connected.
Make sure you are fully addressing the technical
definitions involved --- do not just talk vaguely about
vertices being connected, you need to discuss specific paths
between vertices.

Suppose that H is a connected graph that contains a
proper cycle. Let H′ represent the subgraph of H that results by
removing a single edge from H, where the edge removed is part of
the proper cycle that H contains. Argue that H′ remains
connected.
Notes.
Your argument here needs to be (slightly) different from your
argument in Activity 16.3.
Make sure you are using the technical definition of connected
graph in your argument. What are you assuming about...

(a) Let L be a minimum edge-cut in a connected graph G with at
least two vertices. Prove that G − L has exactly two
components.
(b) Let G an eulerian graph. Prove that λ(G) is even.

10.-Construct a connected bipartite graph that is not a tree
with vertices Q,R,S,T,U,V,W.
What is the edge set?
Construct a bipartite graph with vertices Q,R,S,T,U,V,W such
that the degree of S is 4.
What is the edge set?
12.-Construct a simple graph with vertices F,G,H,I,J that has an
Euler trail, the degree of F is 1 and the degree of G is 3.
What is the edge set?
13.-Construct a simple graph with vertices L,M,N,O,P,Q that has
an Euler circuit...

Let G be a connected simple graph with n vertices and m edges.
Prove that G contains at least m−n+ 1 different subgraphs
which are polygons (=circuits). Note: Different polygons
can have edges in common. For instance, a square with a diagonal
edge has three different polygons (the square and two different
triangles) even though every pair of polygons have at least one
edge in common.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 21 minutes ago

asked 23 minutes ago

asked 26 minutes ago

asked 32 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago