Prove or disprove: If G = (V; E) is an undirected graph where
every vertex has...
Prove or disprove: If G = (V; E) is an undirected graph where
every vertex has degree at least 4 and u is in V , then there are
at least 64 distinct paths in G that start at u.
Given an undirected graph G=(V, E) with weights and a vertex ,
we ask for a minimum weight spanning tree in G where is not a leaf
(a leaf node has degree one). Can you solve this problem in
polynomial time? Please write the proof.
Let G(V, E,w) be a weighted undirected graph, where V is the set
of vertices, E is the set of edges, and w : E → R + is the weight
of the edges (R + is the set of real positive numbers). Suppose
T(G) is the set of all minimum spanning trees of G and is
non-empty. If we know that the weight function w is a injection,
i.e., no two edges in G have the same weight, then:...
Consider an unweighted, undirected graph G = <V, E>. The
neighbourhood of a node v ∈ V in the graph is the set of all nodes
that are adjacent (or directly connected) to v. Subsequently, we
can define the neighbourhood degree of the node v as the sum of the
degrees of all its neighbours (those nodes that are directly
connects to v).
(a) Design an algorithm that returns a list containing the
neighbourhood degree for each node v ∈...
You are given an undirected graph G = ( V, E ) in which the edge
weights are highly restricted. In particular, each edge has a
positive integer weight from 1 to W, where W is a constant
(independent of the number of edges or vertices). Show that it is
possible to compute the single-source shortest paths in such a
graph in O(E+V) time.
Given an undirected graph G = (V,E),
consisting of n vertices and m edges, with each
edge labeled from the set {0,1}.
Describe and analyze the worst-case time complexity of an
efficient algorithm to find any cycle consisting of edges whose
labels alternate 0,1.
Given a connected graph G where edge costs are pair-wise
distinct, prove or disprove that the G has a unique MST.
Please write Pseudo-code for the algorithms.