5.1: Discovering Graphs
Definitions are typically constructed after we work with new objects. The definition is constructed to match the properties we have observed and need. In this section we will practice this by developing a definition for a type of object known as a graph .
Checkpoint \(\PageIndex{3}\)
Based on the examples in Figure \(\PageIndex{1}\) and Figure \(\PageIndex{2}\) determine which of the following are graphs.
Each of these is a graph.
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\begin{align*} P & =\{p_1,p_2,p_3\}\\ Q & =\{q_1,q_2,q_3\}\\ q_1 & = \{p_1,p_2\}\\ q_2 & = \{p_1,p_3\}\\ q_3 & = \{p_2,p_3\} \end{align*}
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\begin{align*} P & =\{p_1,p_2,p_3,p_4\}\\ Q & =\emptyset \end{align*}
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\begin{align*} P & =\{p_1,p_2,p_3,p_4\}\\ Q & =\{q_1,q_2\}\\ q_1 &= \{p_1,p_3\}\\ q_2 & = \{p_2,p_4\} \end{align*}
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\begin{align*} P & =\{p_1,p_2,p_3,p_4,p_5\}\\ L & =\{\ell_1,\ell_2,\ell_3,\ell_4,\ell_5,\ell_6\}\\ \ell_1 & = \{p_2,p_3\}\\ \ell_2 & = \{p_1,p_4\}\\ \ell_3 & = \{p_4,p_5\}\\ \ell_4 & = \{p_1,p_5\}\\ \ell_5 & = \{p_2,p_4\}\\ \ell_6 & = \{p_2,p_5\} \end{align*}
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\begin{align*} V & =\{c,d,e,f\}\\ E & =\{\{c,d\}, \{d,e\}, \{e,f\},\\ & \{c,f\}\} \end{align*}
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\begin{align*} V & =\{v_1,v_2,v_3,v_4\}\\ E & =\{\{v_1,v_2\}, \{v_1,v_3\}, \\ & \{v_2,v_3\}, \{v_2,v_4\}\} \end{align*}
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\begin{align*} P & =\{p\}\\ Q & =\emptyset \end{align*}
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\begin{align*} P & =\{q,r,s,t\}\\ Q &=\{\{q,r\}, \{r,t\}, \{q,t\}, \{r,s\}\} \end{align*}
None of these is a graph.
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\begin{align*} P & =\{p_1,p_2,p_3\}\\ Q & =\{q_1,q_2\}\\ q_1 & = \{p_1,p_2\}\\ q_2 & = \{p_3,p_4\} \end{align*}
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\begin{align*} P & =\{p_1,p_2\}\\ Q & =\{q_1,q_2\}\\ q_1 & = \{p_1,p_2\}\\ q_2 & = \{p_1,p_2\} \end{align*}
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\begin{align*} P & =\{p_1,p_2,p_3\}\\ Q & =\{q_1,q_2,q_3\}\\ q_1 \&= (p_1,p_2)\\ q_2 & = (p_1,p_3)\\ q_3 & = (p_2,p_3) \end{align*}
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\begin{align*} V & =\{x,y,z\}\\ E & =\{f,g,h\}\\ f & = (x,y)\\ g & = (y,x)\\ h & = (y,z) \end{align*}
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\begin{align*} P & =\emptyset\\ Q & =\emptyset \end{align*}
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\begin{align*} P \& =\{q,r,s,t\}\\ Q & =\{\{q,r\}, \{r,t\}, \{s\} \} \end{align*}
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\begin{align*} V & =\{p_1,p_2,\ldots,p_n,\ldots\}\\ E & =\{\{p_1,p_2\}, \{p_2,p_4\}, \{p_1,p_3\}, \{p_4,p_6\}\} \end{align*}
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\begin{align*} P & =\{p_1,p_2,p_3\}\\ Q & =\{q_1,q_2,q_3\}\\ q_1 & = \{p_1,p_2\}\\ q_2 & = \{p_1,p_3\} \end{align*}
Checkpoint \(\PageIndex{6}\)
Based on the examples in Example 5.1.4 and Example 5.1.5 determine which of the following are graphs.
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\begin{align*} P & =\{p_1,p_2,p_3\}\\ Q & =\{q_1\}\\ q_1 & = \{p_1,p_3\} \end{align*}
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\begin{align*} P & =\{p_1,p_2,p_3\}\\ Q & =\{q_1,q_2\}\\ q_1 & = \{p_1,p_2\} \end{align*}
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\begin{align*} V & =\{v_1,v_2,\ldots,v_n,\ldots\}\\ E & =\{e_1,e_2,\ldots,e_n,\ldots\}\\ e_1 & = (v_1,v_2)\\ \vdots\\ e_n & = (v_n,v_{n+1})\\ \vdots \end{align*}
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\begin{align*} P & =\{p,q,r\}\\ Q & =\{t,u,v\}\\ t & = \{p,q\}\\ u & = \{p,r\}\\ v & = \{q,r\} \end{align*}
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\begin{align*} P & =\{p_1,p_2,p_3\}\\ Q & =\emptyset \end{align*}
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\begin{align*} V & =\{u_1,u_2,u_3,u_4\}\\ E & =\{(u_1,u_2), (u_2,u_1),\\ & (u_3,u_4)\} \end{align*}
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\begin{align*} V & =\{v_1,v_2,v_3\}\\ E & =\{\{v_1,v_2\}, \{v_1,v_3\}, \{v_3\}\} \end{align*}
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\begin{align*} P & =\{p,q,r,s\}\\ Q & =\{\{p,q\}, \{p,r\}, \{p,s\},\\ & \{q,r\}, \{q,s\}, \{r,s\}\} \end{align*}