# Polynomially related lengths under two different encodings

I'm reading through "Computers and Intractability: A guide to the Theory of NP-Completeness" by Michael R. Garey and David S. Johnson, p. 20 and I came across this concept of a function that is polynomially related to input lengths obtained using some encoding scheme. Let $$Len:D_{\Pi}\rightarrow \mathbb Z^+$$ be a function that maps instances $\in D_{\Pi}$ (the set of instances of decision problem $\Pi$) to positive integers (lengths). Let $x$ be the string obtained from $I\in D_{\Pi}$ under some encoding $e$. If there exist polynomials $p$ and $p'$ such that $$Len(I) \le p(|x|)$$ and $$|x| \le p'(Len(I)),$$ We say that $Len$ is polynomially related to the input lengths obtained by the encoding $e$. I cannot digest that; my understanding is that two encodings are polynomially related if converting from one another requires a polynomial amount of time. Can anybody clarify things a bit?

Garey and Johnson are referring to the fact that any encoding scheme for some instance $I$ of a problem $\Pi$ will only differ in length (i.e. number of bits) by a polynomial amount. For example, consider two possible ways to encode a graph: adjacency matrix, and adjacency list.