Subdivision of graphs in $R(m{K}_2,P_4)$

For any graphs $F,G$ , and H, the notation $F\to (G,H)$ means that any red-blue coloring of all edges of F will contain either a red copy of G or a blue copy of H. The set $R(G,H)$ consists of all Ramsey $(G,H)$ -minimal graphs, namely all graphs F satisfying $F\to (G,H)$ but for each $e\in E\left(F\right)$ , $(F-e)\nrightarrow (G,H)$ . In this paper, we propose a simple construction for creating new Ramsey minimal graphs from the previous known Ramsey minimal graphs (by subdivision operation). In particular, suppose $F\in R(m{K}_2,P_4)$ and let $e\in E\left(F\right)$ be an edge contained in a cycle of F, we construct a new Ramsey minimal graph in $R\left(\right(m+1)K_2,P_4)$ from graph F by subdividing the edge e four times.