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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) Suppose that V(G)| > 1. Show that G has at least 2 leaves. (You may use the fact proved in class that G has a vertex of degree 1.) (b) Suppose that G has exactly two leaves, prove that G is a path. (c) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (d) Prove that either there is a vertex v € V(G) such that f(v) = v or there is an edge {r, 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) Suppose that V(G)| > 1. Show that G has at least 2 leaves. (You may use the fact proved in class that G has a vertex of degree 1.) (b) Suppose that G has exactly two leaves, prove that G is a path. (c) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (d) Prove that either there is a vertex v € V(G) such that f(v) = v or there is an edge {r, 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.)

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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b,c,d

1. Let G be a tree and let L be the set of leaves in G (the vertices of degree 1). (a) Suppose that V(G)| > 1. Show that G has at least 2 leaves. (You may use the fact proved in class that G has a vertex of degree 1.) (b) Suppose that G has exactly two leaves, prove that G is a path. (c) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (d) Prove that either there is a vertex v € V(G) such that f(v) = v or there is an edge {r, 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.)
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) Suppose that V(G)| > 1. Show that G has at least 2 leaves. (You may use the fact proved in class that G has a vertex of degree 1.) (b) Suppose that G has exactly two leaves, prove that G is a path. (c) Let f be a graph isomorphism from G to G. Prove that f(L) = L. (d) Prove that either there is a vertex v € V(G) such that f(v) = v or there is an edge {r, 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.)
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