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(d) Show that no line graph can contain the star K₁ as an induced subgraph.

(d) Show that no line graph can contain the star K₁ as an induced subgraph.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Hi please assist with Part (d) only . Thank you !

Question 4 Let G(V, E) be a simple graph with minimum degree (G) ≥ 1. The line graph of G, denoted by L(G), has vertex set V₂ = {e: e E E). Two vertices i,j € V, are adjacent if and only if the corresponding edges i, je E are incident to a common vertex. Example: (a) (c) a (d) b C G d Graph H a g b Graph G Graph L(G) Draw the line graphs L(H) and L(K) of graphs H and K respectively. (b) Let G be a simple r-regular graph with r 2 2. Is C(G) also a regular graph? If C(G) Is regular graph, what is the common degree in it? Graph K Characterise, with justification, all simple graphs G such that G and its line graph C(G) are isomorphic. That is, describe, with justification, all simple graphs G such that G and L(G) are isomorphic. Show that no line graph can contain the star K₁3 as an induced subgraph.
Transcribed Image Text:Question 4 Let G(V, E) be a simple graph with minimum degree (G) ≥ 1. The line graph of G, denoted by L(G), has vertex set V₂ = {e: e E E). Two vertices i,j € V, are adjacent if and only if the corresponding edges i, je E are incident to a common vertex. Example: (a) (c) a (d) b C G d Graph H a g b Graph G Graph L(G) Draw the line graphs L(H) and L(K) of graphs H and K respectively. (b) Let G be a simple r-regular graph with r 2 2. Is C(G) also a regular graph? If C(G) Is regular graph, what is the common degree in it? Graph K Characterise, with justification, all simple graphs G such that G and its line graph C(G) are isomorphic. That is, describe, with justification, all simple graphs G such that G and L(G) are isomorphic. Show that no line graph can contain the star K₁3 as an induced subgraph.
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