Asian Research Journal of Mathematics

This study examines the convex-fair domination parameter under four unary graph operations and structural transformations applied to path graphs P_n and cycle graphs C_n. The operations considered are subdivision, vertex duplication, total-graph construction, and vertex switching. Convex-fair domination requires a dominating set to be geodesically convex while also ensuring that every vertex outside the set has the same number of neighbours within it. Using the baseline convexfair domination values stated for paths and cycles, the manuscript derives operation-specific expressions for the resulting transformed graphs. For paths, the analysis treats the subdivision graph S(P_n), duplication graph D(P_n), total graph T (P_n), and the graph obtained by switching a pendant vertex. Corresponding results are developed for cycles through S(C_n), D(C_n), T (C_n), and single-vertex switching. The proofs use structural isomorphisms, explicit candidate dominating sets, geodesicclosure arguments, and comparisons of neighbourhood intersections for vertices outside the selected sets. Attention is also given to how transformed graph order and the placement of boundary and internal vertices affect the proposed minimum sets. The analysis therefore describes how subdivision, duplication, total-graph construction, and switching alter the interaction between domination, convexity, and fairness in linear and cyclic graph structures. These results provide a focused basis for further study of convex-fair domination under other graph operations and in broader graph classes.