# 4.5: Introduction to Graph Theory

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We can interpret the sdr problem as a problem about graphs. Given sets $$A_1,A_2,\ldots,A_n$$, with $$\bigcup_{i=1}^n A_i=\{x_1,x_2,\ldots,x_m\}$$, we define a graph with $$n+m$$ vertices as follows: The vertices are labeled $$\{A_1,A_2,\ldots,A_n,x_1,x_2,\ldots x_m\}$$, and the edges are $$\{\{A_i,x_j\}\mid x_j\in A_i\}$$.

##### Example $$\PageIndex{1}$$

Let $$A_1=\{a,b,c,d\}$$, $$A_2=\{a,c,e,g\}$$, $$A_3=\{b,d\}$$, and $$A_4=\{a,f,g\}$$. The corresponding graph is shown in Figure $$\PageIndex{1}$$.

Before exploring this idea, we introduce a few basic concepts about graphs. If two vertices in a graph are connected by an edge, we say the vertices are adjacent. If a vertex $$v$$ is an endpoint of edge $$e$$, we say they are incident. The set of vertices adjacent to $$v$$ is called the neighborhood of $$v$$, denoted $$N(v)$$. This is sometimes called the open neighborhood of $$v$$ to distinguish it from the closed neighborhood of $$v$$, $$N[v]=N(v)\cup\{v\}$$. The degree of a vertex $$v$$ is the number of edges incident with $$v$$; it is denoted $$\text{d}(v)$$.

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