Threshold graphs under picture Dombi fuzzy information

The aggregation feature, decision-making skills and operational characteristics of multi-purpose Dombi operators make them a highly adaptable tool for compiling the imprecise information. This study exploits the generalized structure of Dombi operators and significant characteristics of picture fuzzy sets \((\mathcal {PFS}_{s})\) to extend the theory of fuzzy graph by presenting the premium concept of picture Dombi fuzzy threshold graphs \((\mathcal {PDFTG}_{s}).\) We prove that \(\mathcal {PDFTG}_{s}\) do not induce picture Dombi fuzzy alternating \((\mathcal {PDFA})\) 4-cycle as induced subgraph, and these graphs can be constructed periodically by adding an isolated or dominant vertex to a single vertex graph. We demonstrate that \(\mathcal {PDFTG}_{s}\) are triangulated graphs. We show that the crisp graph of \(\mathcal {PDFTG}\) is a split graph \(({\mathcal {S}}{\mathcal {G}})\). Further, we illustrate the notion of threshold dimension and threshold partition number of picture Dombi fuzzy graphs \((\mathcal {PDFG}_{s})\). Moreover, we present some fundamental results related to threshold dimension and threshold partition number with the appropriate illustration. Finally, we discuss the implementation of \(\mathcal {PDFTG}_{s}\) in the distribution of coal resources.