We consider partitioned graphs, by which we mean finite directed graphs with a partitioned edge set ${\mathcal{E}}={\mathcal{E}}^{-}\cup {\mathcal{E}}^{+}$ . Additionally given a relation ${\mathcal{R}}$ between the edges in ${\mathcal{E}}^{-}$ and the edges in ${\mathcal{E}}^{+}$ , and under the appropriate assumptions on ${\mathcal{E}}^{-},{\mathcal{E}}^{+}$ and ${\mathcal{R}}$ , denoting the vertex set of the graph by $\mathfrak{P}$ , we speak of an ${\mathcal{R}}$ -graph ${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$ . From ${\mathcal{R}}$ -graphs ${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$ we construct semigroups (with zero) ${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$ that we call ${\mathcal{R}}$ -graph semigroups. We write a list of conditions on a topologically transitive subshift with property $(A)$ that together are sufficient for the subshift to have an ${\mathcal{R}}$ -graph semigroup as its associated semigroup.

Generalizing previous constructions, we describe a method of presenting subshifts by means of suitably structured finite labeled directed graphs $({\mathcal{V}},~\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706}~)$ with vertex set ${\mathcal{V}}$ , edge set $\unicode[STIX]{x1D6F4}$ , and a label map that assigns to the edges in $\unicode[STIX]{x1D6F4}$ labels in an ${\mathcal{R}}$ -graph semigroup ${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$ . We denote the presented subshift by $X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$ and call $X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$ an ${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$ -presentation.

We introduce a property $(B)$ of subshifts that describes a relationship between contexts of admissible words of a subshift, and we introduce a property $(c)$ of subshifts that in addition describes a relationship between the past and future contexts and the context of admissible words of a subshift. Property $(B)$ and the simultaneous occurrence of properties $(B)$ and $(c)$ are invariants of topological conjugacy.

We consider subshifts in which every admissible word has a future context that is compatible with its entire past context. Such subshifts we call right instantaneous. We introduce a property $RI$ of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a right instantaneous presentation. We consider also subshifts in which every admissible word has a future context that is compatible with its entire past context, and also a past context that is compatible with its entire future context. Such subshifts we call bi-instantaneous. We introduce a property $BI$ of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a bi-instantaneous presentation.

We define a subshift as strongly bi-instantaneous if it has for every sufficiently long admissible word $a$ an admissible word $c$ , that is contained in both the future context of $a$ and the past context of $a$ , and that is such that the word $ca$ is a word in the future context of $a$ that is compatible with the entire past context of $a$ , and the word $ac$ is a word in the past context of $a$ , that is compatible with the entire future context of $a$ . We show that a topologically transitive subshift with property $(A)$ , and associated semigroup a graph inverse semigroup ${\mathcal{S}}$ , has an ${\mathcal{S}}$ -presentation, if and only if it has properties $(c)$ and $BI$ , and a strongly bi-instantaneous presentation, if and only if it has properties $(c)$ and $BI$ , and all of its bi-instantaneous presentations are strongly bi-instantaneous.

We construct a class of subshifts with property $(A)$ , to which certain graph inverse semigroups ${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$ are associated, that do not have ${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$ -presentations.

We associate to the labeled directed graphs $({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$ topological Markov chains and Markov codes, and we derive an expression for the zeta function of $X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$ in terms of the zeta functions of the topological Markov shifts and the generating functions of the Markov codes.