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Section 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.
Figure 5.1.1. Each of these is a graph
Figure 5.1.2. None of these is a graph

Example 5.1.4.

Each of these is a graph.
  1. \begin{align*} P \amp =\{p_1,p_2,p_3\}\\ Q \amp =\{q_1,q_2,q_3\}\\ q_1 \amp = \{p_1,p_2\}\\ q_2 \amp = \{p_1,p_3\}\\ q_3 \amp = \{p_2,p_3\} \end{align*}
  2. \begin{align*} P \amp =\{p_1,p_2,p_3,p_4\}\\ Q \amp =\emptyset \end{align*}
  3. \begin{align*} P \amp =\{p_1,p_2,p_3,p_4\}\\ Q \amp =\{q_1,q_2\}\\ q_1 \amp = \{p_1,p_3\}\\ q_2 \amp = \{p_2,p_4\} \end{align*}
  4. \begin{align*} P \amp =\{p_1,p_2,p_3,p_4,p_5\}\\ L \amp =\{\ell_1,\ell_2,\ell_3,\ell_4,\ell_5,\ell_6\}\\ \ell_1 \amp = \{p_2,p_3\}\\ \ell_2 \amp = \{p_1,p_4\}\\ \ell_3 \amp = \{p_4,p_5\}\\ \ell_4 \amp = \{p_1,p_5\}\\ \ell_5 \amp = \{p_2,p_4\}\\ \ell_6 \amp = \{p_2,p_5\} \end{align*}
  5. \begin{align*} V \amp =\{c,d,e,f\}\\ E \amp =\{\{c,d\}, \{d,e\}, \{e,f\},\\ \amp \{c,f\}\} \end{align*}
  6. \begin{align*} V \amp =\{v_1,v_2,v_3,v_4\}\\ E \amp =\{\{v_1,v_2\}, \{v_1,v_3\}, \\ \amp \{v_2,v_3\}, \{v_2,v_4\}\} \end{align*}
  7. \begin{align*} P \amp =\{p\}\\ Q \amp =\emptyset \end{align*}
  8. \begin{align*} P \amp =\{q,r,s,t\}\\ Q \amp =\{\{q,r\}, \{r,t\}, \{q,t\}, \{r,s\}\} \end{align*}

Example 5.1.5.

None of these is a graph.
  1. \begin{align*} P \amp =\{p_1,p_2,p_3\}\\ Q \amp =\{q_1,q_2\}\\ q_1 \amp = \{p_1,p_2\}\\ q_2 \amp = \{p_3,p_4\} \end{align*}
  2. \begin{align*} P \amp =\{p_1,p_2\}\\ Q \amp =\{q_1,q_2\}\\ q_1 \amp = \{p_1,p_2\}\\ q_2 \amp = \{p_1,p_2\} \end{align*}
  3. \begin{align*} P \amp =\{p_1,p_2,p_3\}\\ Q \amp =\{q_1,q_2,q_3\}\\ q_1 \amp = (p_1,p_2)\\ q_2 \amp = (p_1,p_3)\\ q_3 \amp = (p_2,p_3) \end{align*}
  4. \begin{align*} V \amp =\{x,y,z\}\\ E \amp =\{f,g,h\}\\ f \amp = (x,y)\\ g \amp = (y,x)\\ h \amp = (y,z) \end{align*}
  5. \begin{align*} P \amp =\emptyset\\ Q \amp =\emptyset \end{align*}
  6. \begin{align*} P \amp =\{q,r,s,t\}\\ Q \amp =\{\{q,r\}, \{r,t\}, \{s\} \} \end{align*}
  7. \begin{align*} V \amp =\{p_1,p_2,\ldots,p_n,\ldots\}\\ E \amp =\{\{p_1,p_2\}, \{p_2,p_4\}, \{p_1,p_3\}, \{p_4,p_6\}\} \end{align*}
  8. \begin{align*} P \amp =\{p_1,p_2,p_3\}\\ Q \amp =\{q_1,q_2,q_3\}\\ q_1 \amp = \{p_1,p_2\}\\ q_2 \amp = \{p_1,p_3\} \end{align*}

Checkpoint 5.1.6.

Based on the examples in ExampleΒ 5.1.4 and ExampleΒ 5.1.5 determine which of the following are graphs.
  1. \begin{align*} P \amp =\{p_1,p_2,p_3\}\\ Q \amp =\{q_1\}\\ q_1 \amp = \{p_1,p_3\} \end{align*}
  2. \begin{align*} P \amp =\{p_1,p_2,p_3\}\\ Q \amp =\{q_1,q_2\}\\ q_1 \amp = \{p_1,p_2\} \end{align*}
  3. \begin{align*} V \amp =\{v_1,v_2,\ldots,v_n,\ldots\}\\ E \amp =\{e_1,e_2,\ldots,e_n,\ldots\}\\ e_1 \amp = (v_1,v_2)\\ \vdots\\ e_n \amp = (v_n,v_{n+1})\\ \vdots \end{align*}
  4. \begin{align*} P \amp =\{p,q,r\}\\ Q \amp =\{t,u,v\}\\ t \amp = \{p,q\}\\ u \amp = \{p,r\}\\ v \amp = \{q,r\} \end{align*}
  5. \begin{align*} P \amp =\{p_1,p_2,p_3\}\\ Q \amp =\emptyset \end{align*}
  6. \begin{align*} V \amp =\{u_1,u_2,u_3,u_4\}\\ E \amp =\{(u_1,u_2), (u_2,u_1),\\ \amp (u_3,u_4)\} \end{align*}
  7. \begin{align*} V \amp =\{v_1,v_2,v_3\}\\ E \amp =\{\{v_1,v_2\}, \{v_1,v_3\}, \{v_3\}\} \end{align*}
  8. \begin{align*} P \amp =\{p,q,r,s\}\\ Q \amp =\{\{p,q\}, \{p,r\}, \{p,s\},\\ \amp \{q,r\}, \{q,s\}, \{r,s\}\} \end{align*}