Question

Show that an edge e of a connected graph G belongs to any spanning tree of G if and only if e is a bridge of G. Show that e does not belong to any spanning tree if and only if e is a loop of G.

Answer #1

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.)

Let e be the unique lightest edge in a graph G. Let T be a
spanning tree of G such that e ∉ T . Prove using elementary
properties of spanning trees (i.e. not the cut property) that T is
not a minimum spanning tree of G.

Let G be a connected plane graph and let T be a spanning tree of
G. Show that those edges in G∗ that do not correspond to the edges
of T form a spanning tree of G∗ . Hint: Use all you know about
cycles and cutsets!

Proof: Let G be a k-connected k-regular graph. Show that, for
any edge e, G has a perfect matching M such that e ε
M.
Please show full detailed proof. Thank you in advance!

A Hamiltonian walk in a connected graph G is a closed spanning
walk of minimum length in G. PROVE that every connected graph G of
size m contains a Hamiltonian walk of length at most 2m in which
each edge of G appears at most twice.

Show Proof of correctness and state, and solve the Recurrence
using the Master Theorem. Let G = G(V, E) be an arbitrary,
connected, undirected graph with vertex set V and edge set E.
Assume that every edge in E represents either a road or a bridge.
Give an efficient algorithm that takes as input G and decides
whether there exists a spanning tree of G where the number of edges
that represents roads is
floor[ (|V|/ √ 2) ]. Do...

Consider edges that must be in every spanning tree of a graph.
Must every graph have such an edge? Give an example of a graph that
has exactly one such edge.

Consider an undirected graph G = (V, E) with an injective cost
function c: E → N. Suppose T is a minimum spanning tree of G for
cost function c. If we replace each edge cost c(e), e ∈ E, with
cost c'(e) = c(e)2 for G, is T still a minimum spanning
tree of G? Briefly justify your answer.

Let T be a minimum spanning tree of graph G obtained by Prim’s
algorithm. Let Gnew be a graph obtained by adding to G a new vertex
and some edges, with weights, connecting the new vertex to some
vertices in G. Can we construct a minimum spanning tree of Gnew by
adding one of the new edges to T ? If you answer yes, explain how;
if you answer no, explain why not.

Let G be a connected graph. Show that G has a subtree that is a
maximal tree.

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