Let G be a graph with vertex set V. Define a
relation R from V to...
Let G be a graph with vertex set V. Define a
relation R from V to itself as follows: vertex
u has this relation R with vertex v,
u R v, if there is a path in G from u to
v. Prove that this relation is an equivalence relation.
Write your proof with complete sentences line by line in a logical
order. If you can, you may write your answer to this
question directly in the space provided.Your presentation
counts.
Let G = (V,E) be a graph with n vertices and e edges. Show that
the...
Let G = (V,E) be a graph with n vertices and e edges. Show that
the following statements are equivalent:
1. G is a tree
2. G is connected and n = e + 1
3. G has no cycles and n = e + 1
4. If u and v are vertices in G, then there exists a unique path
connecting u and v.
You are given a directed acyclic graph G(V,E), where each vertex
v that has in-degree 0...
You are given a directed acyclic graph G(V,E), where each vertex
v that has in-degree 0 has a value value(v) associated with it. For
every other vertex u in V, define Pred(u) to be the set of vertices
that have incoming edges to u. We now define value(u) = ?v∈P red(u)
value(v). Design an O(n + m) time algorithm to compute value(u) for
all vertices u where n denotes the number of vertices and m denotes
the number of edges...
A K-regular graph G is a graph such that deg(v) = K for all
vertices v...
A K-regular graph G is a graph such that deg(v) = K for all
vertices v in G. For example, c_9 is a 2-regular graph, because
every vertex has degree 2. For some K greater than or equal to 2,
neatly draw a simple K-regular graph that has a bridge. If it is
impossible, prove why.
Let G = (X, E) be a connected graph. The distance between two
vertices x and...
Let G = (X, E) be a connected graph. The distance between two
vertices x and y of G is the shortest length of the paths linking x
and y. This distance is denoted by d(x, y). We call the center of
the graph any vertex x such that the quantity max y∈X d(x, y) is
the smallest possible. Show that if G is a tree then G has either
one center or two centers which are then neighbors
10.-Construct a connected bipartite graph that is not a tree
with vertices Q,R,S,T,U,V,W.
What is the...
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...