Two positive numbers $A$ and $B$ are said to be connected (denoted by "$A\leftrightarrow B$") if one of these conditions holds:
(1) $A$ and $B$ have the same length and differ in exactly one digit; for example, $123\leftrightarrow 173$.
(2) Adding one digit to the left of $A$ (or $B$) makes $B$ (or $A$); for example, $23\leftrightarrow 223$ and $123\leftrightarrow 23$.

We call a prime $P$ a $2$'s relative if there exists a chain of connected primes between $2$ and $P$ and no prime in the chain exceeds $P$.

For example, $127$ is a $2$'s relative. One of the possible chains is shown below:
$2\leftrightarrow 3\leftrightarrow 13\leftrightarrow 113\leftrightarrow 103\leftrightarrow 107\leftrightarrow 127$
However, $11$ and $103$ are not $2$'s relatives.

Let $F(N)$ be the sum of the primes $\le N$ which are not $2$'s relatives.
We can verify that $F({10}^3)=431$ and $F({10}^4)=78728$.

Find $F({10}^7)$.