SOFT DISEMIGRAPHS: EXPLORATION OF CERTAIN EXTENDED AND RESTRICTED OPERATIONS

Soft set theory offers a systematic method for addressing imprecision and uncertainty by classifying set elements based on specific parameters. In the realm of semigraph theory, soft semigraphs employ this method, providing a parametrized perspective that has greatly enhanced the field through efficient parameter management. Building on this foundation, disemigraphs extend semigraphs by incorporating directional relationships among vertices, making them ideal for modeling situations where the sequence and direction of connections are essential. However, existing models such as fuzzy graphs or soft graphs often fall short in capturing both the directionality and parameter-based uncertainty simultaneously, especially in networks involving multiple interaction types and layered relational structures. This paper introduces the concept of soft disemigraphs, a novel framework that integrates the flexibility of soft set theory with the directional rigor of disemigraphs. This integration enables a more nuanced, structured, and context-aware representation of complex systems, particularly where directionality, hierarchy, and imprecision coexist. We explore various operations on soft disemigraphs, including extended union, restricted union, extended intersection, and restricted intersection, and analyse their structural properties. These operations provide deeper insights into the dynamics of parametrized, directed networks and offer a clear advantage in modeling real-world systems. Furthermore, we demonstrate the practical applicability of this framework through ecological network modeling, particularly in the representation of food webs, where species interactions are inherently directional, hierarchical, and context-sensitive.