Let T = (V, E) be a tree, and suppose that some node u ∈ V...
Let T = (V, E) be a tree, and suppose that some node u ∈ V has
degree d. Prove that T has at least d leaves.
Hint: Consider the induced subgraph with vertex set V \ {u}
Let G = (V, E) be a tree, and let M be the greatest possible
number...
Let G = (V, E) be a tree, and let M be the greatest possible
number of vertices in a path that is a subgraph of G. Show that any
two paths with M vertices in G must have at least one vertex in
common.
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.
A spanning tree of connected graph G = (V, E) is an acyclic
connected subgraph (V,...
A spanning tree of connected graph G = (V, E) is an acyclic
connected subgraph (V, E0 ) with the same vertices as G. Show that
every connected graph G = (V, E) contains a spanning tree. (It is
the connected subgraph (V, E0 ) with the smallest number of
edges.)
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...
Let G=(V,E) be a connected graph with |V|≥2
Prove that ∀e∈E the graph G∖e=(V,E∖{e}) is disconnected,...
Let G=(V,E) be a connected graph with |V|≥2
Prove that ∀e∈E the graph G∖e=(V,E∖{e}) is disconnected, then G
is a tree.
Prove that if G = (V, E) is a tree and e ∈ E, then (V,...
Prove that if G = (V, E) is a tree and e ∈ E, then (V, E − {e})
is a forest of two trees.
Let T be a complete binary tree such that node v stores
the entry (p(v), 0),...
Let T be a complete binary tree such that node v stores
the entry (p(v), 0), where p(v) is the level number of v. Is tree T
a heap? Why or why not?
I know that a complete binary tree is a heap, but shouldn't we
also take into consideration the values that it is storing into the
tree: (p(v), 0)? The heap tree could be either a min-heap or
max-heap. If we order the the value based of p(v)...
Let T be a tree of order at least 4, and let e1, e2, e3 ∈...
Let T be a tree of order at least 4, and let e1, e2, e3 ∈ E(T¯)
(compliment of T).
Prove that T + e1 + e2 + e3 is planar.