On words that nearly $θ$-commute

The Hamming distance between two equal length words $\alpha, \beta$ is the number of positions where $\alpha$ and $\beta$ differ. For $x, y \in \Sigma^*$ and antimorphic involution $\theta$, $x$ $\theta$-commutes with $y$, if the Hamming distance between $xy$ and $\theta(y)x$ is zero. When the Hamming distance between $xy$ and $\theta(y)x$ attains its minimum non-zero value of one, then we say $x$ nearly $\theta$-commutes with $y$. This manuscript investigates properties of $x$ and $y$ such that the Hamming distance between $xy$ and $\theta(y)x$ is one. We provide a complete characterization of such $x$ and $y$. We introduce a binary relation $R_\theta$ on $\Sigma^*$, where $x R_\theta y$ holds if and only if $x$ nearly $\theta$-commutes with $y$. Finally, for a given $y$, we collect all $x$ such that $x$ nearly $\theta$-commutes with $y$, and discuss various properties of this set.