On characterization of functions preserving metric-type conditions via triangular and polygonal structures

Following the train of thought from our previous paper we revisit the theorems of Pongsriiam and Termwuttipong by further developing their characterization of certain property-preserving functions using the so-called triangle triplets. We develop more general analogues of disjoint sum lemmas for broader classes of metric-type spaces and we apply these to extend results of Bors\`ik an Dobo\v{s} as well as those obtained by Khemaratchatakumthorn and Pongsriiam. As a byproduct we obtain methods of generating non-trivial and infinite strong $b$-metric spaces which are not metric.