In order to extend the algorithms to handle more general classes of forbidden subgraphs beyond Kt+1, Kt,t, and Kp^q, we can modify the construction of gadgets and the overall approach used in the algorithms.
Generalized Gadgets: Instead of designing gadgets specifically for Kt+1, Kt,t, and Kp^q, we can create gadgets that can accommodate a wider range of forbidden subgraphs. These gadgets should be flexible enough to represent various types of forbidden subgraphs based on their structural characteristics.
Dynamic Construction: The algorithms can be designed to dynamically construct gadgets based on the properties of the forbidden subgraphs present in the input graph. This adaptive approach would allow the algorithms to handle a broader spectrum of forbidden subgraphs without the need for predefined gadget structures.
Enhanced Matching Criteria: The matching criteria used in the algorithms can be generalized to consider the specific characteristics of different types of forbidden subgraphs. By incorporating more flexible matching conditions, the algorithms can effectively handle a diverse set of forbidden subgraphs.
Advanced Weight Assignment: The weight assignment mechanism can be enhanced to accommodate a wider range of weight functions that are vertex-induced on various types of forbidden subgraphs. This flexibility in weight assignment would enable the algorithms to address the optimization requirements of different forbidden subgraphs.