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1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L|> k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v € V(G) such that f(v) = v or there is an edge {x,y} = E(G) such that {f(x), f(y)} = {x,y}. (Hint: Induction on V(G); in the induction step consider a restriction of f to a subset of vertices.)

1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L|> k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v € V(G) such that f(v) = v or there is an edge {x,y} = E(G) such that {f(x), f(y)} = {x,y}. (Hint: Induction on V(G); in the induction step consider a restriction of f to a subset of vertices.)

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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1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L| > k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v e V(G) such that f(v) = v or there is an edge {x, y} € E(G) such that {f(x), f(y)} = {x, y}. (Hint: Induction on |V(G)|; in the induction step consider a restriction of ƒ to a subset of vertices.)
Transcribed Image Text:1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) If G contains a vertex of degree k, show that |L| > k. (b) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (c) Prove that either there is a vertex v e V(G) such that f(v) = v or there is an edge {x, y} € E(G) such that {f(x), f(y)} = {x, y}. (Hint: Induction on |V(G)|; in the induction step consider a restriction of ƒ to a subset of vertices.)
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