How to find all extremal graphs using symmetric subgraphs

Let F be a finite family of graphs with minF∈ Fχ(F)=r+1≥3, where $χ(F)$ is the chromatic number of F. Set t=maxF|F|. Let EX(n,F) be the set of graphs with maximum edges among all the graphs of order n without any F as a subgraph. Let $T(n,r)$ be the Turán graph of order n with r parts. Assume that some F₀ is a subgraph of the graph obtained from $T(rt,r)$ by embedding a path in its one part. Simonovits {S1} introduced the concept of symmetric subgraphs, and proved that there exist graphs in EX(n,F) which have symmetrical property. In this paper, we aim to find a way to characterize all the extremal graphs for such F using symmetric subgraphs. Some new extremal results are obtained.