This paper presents a reformulation of statistical thermodynamics for fluids of molecules with highly directional attractive forces. The molecular model includes a repulsive core and multiple short-ranged attraction sites. The authors explore the relationship between graph cancellation in the fugacity expansion and three types of steric incompatibility between repulsive and attractive interactions. Steric effects are used to control density parameters when articulation points are eliminated in the graphical representation. Each density parameter corresponds to a singlet density for a species with a specified set of bonded sites. These densities satisfy subsidiary conditions of internal consistency, equivalent to minimizing the Helmholtz free energy A. Graphical expressions for A and pressure p are derived. The paper also finds analogous expressions to the s-particle direct correlation functions and the Ornstein-Zernike equation. The paper discusses the statistical mechanics of fluids with highly directional attractive forces, focusing on the case of molecules with multiple attraction sites. The model includes a repulsive core and several attraction sites. The authors avoid reducing the model to a description where sites replace molecules as primary units. Instead, they consider multiple species of particles, each representing a monomeric unit with a specified set of bonded attraction sites. The decision of whether a site is bonded is based on graph theory, using Mayer f-functions for core-core and site-site potentials. Graphs become a flexible tool for incorporating geometric information of interactions. The paper shows that regrouping in graphs controls density parameters when articulation points are eliminated. This approach incorporates a great deal of physics in going from fugacity graphs to density graphs. The advantages of this approach are lost when reducing to a single density based on the total f-bond, as shown in the analysis of dimer-forming systems. The paper contains the mechanics and physical justification of the reformulation, with possible approximation theories discussed in a companion paper.This paper presents a reformulation of statistical thermodynamics for fluids of molecules with highly directional attractive forces. The molecular model includes a repulsive core and multiple short-ranged attraction sites. The authors explore the relationship between graph cancellation in the fugacity expansion and three types of steric incompatibility between repulsive and attractive interactions. Steric effects are used to control density parameters when articulation points are eliminated in the graphical representation. Each density parameter corresponds to a singlet density for a species with a specified set of bonded sites. These densities satisfy subsidiary conditions of internal consistency, equivalent to minimizing the Helmholtz free energy A. Graphical expressions for A and pressure p are derived. The paper also finds analogous expressions to the s-particle direct correlation functions and the Ornstein-Zernike equation. The paper discusses the statistical mechanics of fluids with highly directional attractive forces, focusing on the case of molecules with multiple attraction sites. The model includes a repulsive core and several attraction sites. The authors avoid reducing the model to a description where sites replace molecules as primary units. Instead, they consider multiple species of particles, each representing a monomeric unit with a specified set of bonded attraction sites. The decision of whether a site is bonded is based on graph theory, using Mayer f-functions for core-core and site-site potentials. Graphs become a flexible tool for incorporating geometric information of interactions. The paper shows that regrouping in graphs controls density parameters when articulation points are eliminated. This approach incorporates a great deal of physics in going from fugacity graphs to density graphs. The advantages of this approach are lost when reducing to a single density based on the total f-bond, as shown in the analysis of dimer-forming systems. The paper contains the mechanics and physical justification of the reformulation, with possible approximation theories discussed in a companion paper.