Isomorphism classes of graph that belong to both Families(of Graph)

Question

Consider the following four families of Graphs:

• $A=\left\{\text{Paths} \right\}$
• $B=\left\{\text{Cycles} \right\}$
• $C=\left\{\text{Complete Graphs} \right\}$
• $D=\left\{\text{Bipartite Graphs} \right\}$

For Each Pair of these families,determine all isomorphism classes of Graphs that belong to both families.

I am Done with other pairs but stuck in the given following pair.They are-:

$A\bigcap B$ $\Rightarrow$ Solution is $\phi$ ,reason given is that Cycle have Equal number of edges and vertices(i agree!) but can't path have such property?

similar doubt with
$A\bigcap D$

Please help me out !!

Answer 1

For $A\cap B$, a path is a special type of tree. A tree of order $n$ has $n$ vertices and $n-1$ edges, which does not satisfy the definition of a cycle ($n$ vertices and $n$ edges).

For $A\cap D$, a path of arbitrary order ($n>1$) can be drawn in a zig-zag way to construct a bipartite graph. But not all bipartite graphs are path. So, the intersection is $A$ itself.