The term "path graph" refers to a specific type of graph in graph theory. A path graph is a graph that consists of a sequence of vertices connected by edges in a linear manner. In other words, it is a graph where all the vertices are arranged in a line, and each vertex is connected to its adjacent vertices by edges.
For example, a path graph with 5 vertices would look like this:
1 -- 2 -- 3 -- 4 -- 5
The question of whether a path graph is planar or non-planar depends on the number of vertices it contains.
A path graph with two vertices is always planar because it consists of only one edge connecting the two vertices, and this edge can be represented on a plane without any crossings.
However, a path graph with three or more vertices is considered non-planar. The reason is that as the number of vertices increases, it becomes impossible to represent the graph on a plane without any edge crossings. This violates one of the necessary conditions for planarity, known as Kuratowski's theorem, which states that a graph is non-planar if and only if it contains a subgraph that is homeomorphic to either a complete graph on five vertices or the complete bipartite graph on three vertices.
In summary, a path graph with two vertices is planar, but a path graph with three or more vertices is non-planar.