Covering and independence in triangle structures

Let G be a graph in which each edge is contained in at least one triangle (complete subgraph on three vertices). We investigate relationships between the smallest cardinality of an edge set containing at least i edges of each triangle and the largest cardinality of an edge set containing at most j edges of each triangle (i, j ε {1,2}), and also compare those invariants with the numbers of vertices and edges in G. Several open problems are raised in the concluding section.