(This question has nothing to do with colorings.) Suppose a graph G has a Hamilton path consisting of vertices x1, x2,..., Xn. Conside any induced subgraph G' formed by removing a single vertex from G (i.e., G' is formed from G by removing a single vertex and all edge with that vertex as an endpoint.) Prove that G' has at most two connected components, i.e., the vertices of G' can be partitioned into two sets V and V, such that there is a walk between any two vertices in V and a walk between any two vertices in V2. (Hint: You can define V and V2 using the Hamilton path. In some cases V or V2 can be empty, in which case G' is connected.)
(This question has nothing to do with colorings.) Suppose a graph G has a Hamilton path consisting of vertices x1, x2,..., Xn. Conside any induced subgraph G' formed by removing a single vertex from G (i.e., G' is formed from G by removing a single vertex and all edge with that vertex as an endpoint.) Prove that G' has at most two connected components, i.e., the vertices of G' can be partitioned into two sets V and V, such that there is a walk between any two vertices in V and a walk between any two vertices in V2. (Hint: You can define V and V2 using the Hamilton path. In some cases V or V2 can be empty, in which case G' is connected.)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Transcribed Image Text:(This question has nothing to do with colorings.) Suppose a graph G has a Hamilton path consisting of vertices x1, x2, ..., xn. Consider
any induced subgraph G' formed by removing a single vertex from G (i.e., G' is formed from G by removing a single vertex and all edges
with that vertex as an endpoint.) Prove that G' has at most two connected components, i.e., the vertices of G' can be partitioned into
two sets Vi and V2 such that there is a walk between any two vertices in Vị and a walk between any two vertices in V2. (Hint: You can
define Vị and V2 using the Hamilton path. In some cases V1 or V2 can be empty, in which case G' is connected.)
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