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Understanding the Complement of a Graph in Math 170

This document explores the concept of the complement of a graph as taught in Math 170 under Dr. Lipika Deka. A simple graph G has a complement G' where the vertex set remains the same but the edges are inverted; edges exist in G' only if they do not exist in G. We work through Problem #39, providing visual representations of the graphs and detailing the complements for two distinct cases: part (a) and part (b). This guide aids in understanding graph theory fundamentals essential for mathematical studies.

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Understanding the Complement of a Graph in Math 170

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  1. Finding The Compliment of a Graph Jeffrey Martinez Math 170 Dr. LipikaDeka 12/19/13

  2. Chapter 10.1 Problem #39

  3. Compliment of a Graph • From our text, we find the definition of “The Compliment” of a graph: • Definition: If G is a simple graph, the complement of G, denoted G′, is obtained as follows: The vertex set of G′ is identical to the vertex set of G. However, two distinct vertices v and w of G′ are connected by an edge if, and only if, v and w are not connected by an edge in G.

  4. Problem #39 part (a) • The Graph of G in part (a) is shown with vertexes from v1 to v2, from v1 to v4, from v2 to v3, and from v2 to v4. • To find the compliment G’, we keep the vertexes the same, but instead connect any vertexes that did not have an edge between them in G. In this case it is from v1 to v3, and from v3 to v4. A rough sketch of G’ is shown below. v2 v3 v1 v4

  5. Problem #39 part (b) • The Graph of G in part (b) is shown with vertexes from v1 to v2, and from v3 to v4. • Once again, like part (a), to find the compliment G’, we keep the vertexes, connecting any vertexes that do not have an edge between them in the graph of G. In this case it is from v1 to v3, from v1 to v4, from v2to v4 and from v2to v3. A rough sketch of G’ is shown below. v2 v1 v3 v4

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