Example 4.1.4.
Consider the graphs:
Here both \(G_2\) and \(G_3\) are subgraphs of \(G_1\text{.}\) But only \(G_2\) is an induced subgraph. Every edge in \(G_1\) that connects vertices in \(G_2\) is also an edge in \(G_2\text{.}\) In \(G_3\text{,}\) the edge \(\{a,b\}\) is in \(E_1\) but not \(E_3\text{,}\) even though vertices \(a\) and \(b\) are in \(V_3\text{.}\)
The graph \(G_4\) is NOT a subgraph of \(G_1\text{,}\) even though it looks like all we did is remove vertex \(e\text{.}\) The reason is that in \(E_4\) we have the edge \(\{c,f\}\) but this is not an element of \(E_1\text{,}\) so we don't have the required \(E_4 \subseteq E_1\text{.}\)