Designing programming languages that enable intuitive and safe manipulation of data structures is a critical research challenge. Conventional destructive memory operations using pointers are complex and prone to errors. Existing type systems, such as affine types and shape types, address this problem towards safe manipulation of heaps and pointers, but design of high-level declarative languages that allow us to manipulate complex pointer data structures at a higher level of abstraction is largely an open problem. The $\lambda_{GT}$ language, a purely functional programming language that treats hypergraphs (hereafter referred to as graphs) as primary data structures, addresses some of these challenges. By abstracting data with shared references and cycles as graphs, it enables declarative operations through pattern matching and leverages its type system to guarantee safety of these operations. Nevertheless, the previously proposed type system of $\lambda_{GT}$ leaves two significant open challenges. First, the type system does not support \emph{incomplete graphs}, that is, graphs in which some elements are missing from the graphs of user-defined types. Second, the type system relies on dynamic type checking during pattern matching. This study addresses these two challenges by incorporating linear implication into the $\lambda_{GT}$ type system, while introducing new constraints to ensure its soundness.

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