Challenges of Fiat-Shamir Transformation of ZK Proof / Sigma Protocol

Let $$\Sigma$$ be a sigma protocol whose commitment, challenge and response phases are $$\Sigma_1, \Sigma_2, \Sigma_3$$, respectively.

In Fiat-Shamir Transformation, the challenges are $$H(\Sigma_1)$$ where $$H$$ is some hash function.

However, in some literature, instead of having $$H(\Sigma_1)$$ it is $$H(x \parallel \Sigma_1)$$, where $$\parallel$$ denotes concatenation and $$x$$ is the common input. For example, if $$\Sigma$$ is the graph 3-coloring protocol, $$x$$ would be the corresponding graph.

1. Why consider the concatenation with $$x$$? Is $$x$$ there in order to prevent malleable attacks?

2. Consider a language $$L \in$$ NP and suppose that a prover P wants to prove to a verifier V that $$x \in L$$. We use the classical reductions of $$x$$ to a SAT instance and then to a G3C instance. Call this reduction $$t$$. In this way, $$x \in L$$ iff $$t(x)$$ is a 3-colorable graph. Now we can apply the Fiat-Shamir Transformation, and the challenges become $$H(t(x) \parallel \Sigma_1)$$. Is it possible to make the challenges $$H(x \parallel \Sigma_1)$$ instead of $$H(t(x) \parallel \Sigma_1)$$, thus avoiding increase of the input of $$H$$?