Intuition on $P_3$ and $P_4$-free graphs

Question

I am struggling to understand the structure of $P_3$ and $P_4$ graphs. Could someone provide me a few examples of graphs from each of these two classes?

Answer 1

First, I will assume for now that your $H$-free graph means a graph that does not contain $H$ as a subgraph. $P_n$ represents a path with $n$ vertices.

In fact, we only need to consider connected graphs since if a graph is not connected, it can be obtained by some connected components. First of all, any graph has a spanning tree. If a graph does not contain $P_3$, it means that its spanning tree can only be $P_2$ or $P_1$. We can easily see that connected $P_3$-free graphcan only be $P_2$ or $P_1$ as well.

Similarly, the length of the longest path in a spanning tree of a $P_4$-free graph cannot exceed 2. Therefore, its spanning tree is isomorphic to a star graph, and any edge cannot be added to the star graph. So connected $P_4$-free graphs are star graphs.

Indeed, there is another type of $H$-free graphs that are defined as not containing H as a induced subgraph. Under this definition, I am not clear whether the problem posed by the OP has some characterizations. Perhaps I will come up with a new question.