An efficient algorithm for the density Turán problem of some unicyclic graphs

Let H = (V(H),E(H)) be a simple connected graph of order n with the vertex set V (H) and the edge set E(H). We consider a blow-up graph G[H]. We are interested in the following problem. We have to decide whether there exists a blow-up graph G[H], with edge densities satisfying special conditions (homogeneous or inhomogeneous), such that the graph H does not appear in a blow-up graph as a transversal. We study this problem for unicyclic graphs H with the cycle C₃. We show an efficient algorithm to decide whether a given set of edge densities ensures the existence of H in the blow-up graph G[H].