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Let G be a graph. Color each of the vertices of G either red or black. Let R be the set of red vertices. If a vertex is red, then it may never become black. However, if the criteria laid out in the following Infection Rule are met, then additional vertices may become red. Infection rule: If • R' is the set of currently red vertices, • C' is a component of G - R', •UER', and • v is the unique black neighbor of u in C, then u can infect v and v becomes red. If after a sufficient number of infections every vertex in G is red, then R is an infection set. Definition 0.1. For each graph G, let the infection number of G, denoted I(G), be the graph parameter such that I(G) = min{|R| : R is an infection set of G}. (3) Let T be a tree. Show that I(T) = 1.

Let G be a graph. Color each of the vertices of G either red or black. Let R be the set of red vertices. If a vertex is red, then it may never become black. However, if the criteria laid out in the following Infection Rule are met, then additional vertices may become red. Infection rule: If • R' is the set of currently red vertices, • C' is a component of G - R', •UER', and • v is the unique black neighbor of u in C, then u can infect v and v becomes red. If after a sufficient number of infections every vertex in G is red, then R is an infection set. Definition 0.1. For each graph G, let the infection number of G, denoted I(G), be the graph parameter such that I(G) = min{|R| : R is an infection set of G}. (3) Let T be a tree. Show that I(T) = 1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let G be a graph. Color each of the vertices of G either red or black. Let R be the set of red vertices. If a vertex is red, then it may never become black. However, if the criteria laid out in the following Infection Rule are met, then additional vertices may become red. Infection rule: If • R' is the set of currently red vertices, . C is a component of G - R', •UER', and • v is the unique black neighbor of u in C, then u can infect v and v becomes red. If after a sufficient number of infections every vertex in G is red, then R is an infection set. Definition 0.1. For each graph G, let the infection number of G, denoted I(G), be the graph parameter such that I(G) = min{|R|: R is an infection set of G}. (3) Let T be a tree. Show that I(T) = 1.
Transcribed Image Text:Let G be a graph. Color each of the vertices of G either red or black. Let R be the set of red vertices. If a vertex is red, then it may never become black. However, if the criteria laid out in the following Infection Rule are met, then additional vertices may become red. Infection rule: If • R' is the set of currently red vertices, . C is a component of G - R', •UER', and • v is the unique black neighbor of u in C, then u can infect v and v becomes red. If after a sufficient number of infections every vertex in G is red, then R is an infection set. Definition 0.1. For each graph G, let the infection number of G, denoted I(G), be the graph parameter such that I(G) = min{|R|: R is an infection set of G}. (3) Let T be a tree. Show that I(T) = 1.
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