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Hierarchical dominating set Given an undirected graph G = problem with multiple layers. (V, E), we consider a variant of the minimum dominating set • A layer 1 dominating set, denoted L1, is the set of vertices of a dominating set of G. • A layer 2 dominating set, denoted L2, is a subset of vertices of V that dominates the vertices in L1. • A layer i dominating set, denoted Li, is a subset of vertices of V that dominates the vertices in Li-1. Questions: 1. If G is a path with n nodes, what is the smallest i so that |L¡| =1 ? A proof is expected. Do not hesitate to give only a partial answer (e.g., a lower or an upper bound) if you are unable to find the exact answer.

Hierarchical dominating set Given an undirected graph G = problem with multiple layers. (V, E), we consider a variant of the minimum dominating set • A layer 1 dominating set, denoted L1, is the set of vertices of a dominating set of G. • A layer 2 dominating set, denoted L2, is a subset of vertices of V that dominates the vertices in L1. • A layer i dominating set, denoted Li, is a subset of vertices of V that dominates the vertices in Li-1. Questions: 1. If G is a path with n nodes, what is the smallest i so that |L¡| =1 ? A proof is expected. Do not hesitate to give only a partial answer (e.g., a lower or an upper bound) if you are unable to find the exact answer.

Computer Networking: A Top-Down Approach (7th Edition)
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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Hierarchical dominating set Given an undirected graph G = problem with multiple layers. (V, E), we consider a variant of the minimum dominating set • A layer 1 dominating set, denoted L1, is the set of vertices of a dominating set of G. • A layer 2 dominating set, denoted L2, is a subset of vertices of V that dominates the vertices in L1. • A layer i dominating set, denoted Li, is a subset of vertices of V that dominates the vertices in Li-1. Questions: 1. If G is a path with n nodes, what is the smallest i so that |L¡| =1 ? A proof is expected. %3D Do not hesitate to give only a partial answer (e.g., a lower or an upper bound) if you are unable to find the exact answer. 2. Same question when G is a cycle with n nodes. 3. Give a lower bound on the minimum number k of layers needed on any graph G so that |Lk| = 1. This lower bound may be a formula depending on some parameters of the graphs. Please explain your choice and prove that it is a lower bound.
Transcribed Image Text:Hierarchical dominating set Given an undirected graph G = problem with multiple layers. (V, E), we consider a variant of the minimum dominating set • A layer 1 dominating set, denoted L1, is the set of vertices of a dominating set of G. • A layer 2 dominating set, denoted L2, is a subset of vertices of V that dominates the vertices in L1. • A layer i dominating set, denoted Li, is a subset of vertices of V that dominates the vertices in Li-1. Questions: 1. If G is a path with n nodes, what is the smallest i so that |L¡| =1 ? A proof is expected. %3D Do not hesitate to give only a partial answer (e.g., a lower or an upper bound) if you are unable to find the exact answer. 2. Same question when G is a cycle with n nodes. 3. Give a lower bound on the minimum number k of layers needed on any graph G so that |Lk| = 1. This lower bound may be a formula depending on some parameters of the graphs. Please explain your choice and prove that it is a lower bound.
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