Summary: We develop unifying graph theoretic techniques to study the dynamics and the structure of spaces \(\mathcal H(\{0,1\}^{\mathbb N})\) and \(\mathcal C(\{0,1\}^{\mathbb N})\), the space of homeomorphisms and the space of self-maps of the Cantor space, respectively. Using our methods, we give characterizations which determine when two homeomorphisms of the Cantor space are conjugate to each other. We also give a new characterization of the co-meager conjugacy class of the space \(\mathcal H(\{0,1\}^{\mathbb N})\). The existence of this class was established by Kechris and Rosendal and a specific element of this class was described concretely by Akin, Glasner and Weiss. Our characterization readily implies many old and new dynamical properties of elements of this class. For example, we show that no element of this class has a Li-Yorke pair, implying the well-known Glasner-Weiss result that there is a co-meager subset of \(\mathcal H(\{0,1\}^{\mathbb N})\), each element of which has topological entropy zero. Our analogous investigation in \(\mathcal C(\{0,1\}^{\mathbb N})\) yields a surprising result: there is a co-meager subset of \(\mathcal C(\{0,1\}^{\mathbb N})\) such that any two elements of this set are conjugate to each other by an element of \(\mathcal H(\{0,1\}^{\mathbb N})\). Our description of this class also yields many old and new results concerning the dynamics of a co-meager subset of \(\mathcal C(\{0,1\}^{\mathbb N})\).