Graphs which do not contain $C_4$ as an induced subgraph

Question

I know the definition of subgraph and induced subgraph but I am slightly confused with the following sentence

Graphs that do not contain $C_4$ as an induced subgraph let them call weakly $C_4$-free graphs.

Let's take a look at the following picture:

Graph $G$ contains several copies of $C_4$ as a subgraph, but none of them as an induced subgraph.

I am really confused how to check manually that $G$ does not contain $C_4$ as an induced subgraph.

Can anyone explain it with details, please?

Answer 1

Here is a conceptually easy way to manually check if $G$ contains an induced $C_4$.

Look at all ways to select four an ordered list of distinct vertices $v_1,v_2,v_3,v_4\in V(G)$. If for any of those selections, you have $$ v_1\sim v_2\text{ and }v_2\sim v_3\text{ and }v_3\sim v_4\text{ and }v_4\sim v_1\text{ and }v_1\not\sim v_3\text{ and }v_2\not\sim v_4 $$ then you have found an induced $C_4$. If the above holds for none of the selections, then no induced $C_4$ exists.

The process for checking for a $C_4$ subgraph is the same, except you omit the requirements $v_1\not\sim v_3$ and $v_2\not\sim v_4$.