On C-homogeneous graphs and ordered pencils

Let C be a class of graphs. A definition of a C-homogeneous graph G is given that generalizes that of C-ultrahomogeneous graph by considering each induced subgraph of G in C tethered by an arc. Let 2 < r ∈ Z, σ ∈ (0, r − 1) ∩ Z, t = 2 − 1 and s = 2. We require G to have each edge shared by exactly one copy of the complete subgraph K2s and one of the Turán graph Tts,t. Letting C = {K2s, Tts,t} leads to connected CK2 -homogeneous graphs G, of which there are infinitely many that are not line graphs. They are constructed via ordered pencils and configurations of points and lines, one such configuration per pair (r, σ) as above. Moreover, if r− σ = 2, then G is K4-ultrahomogeneous with order (2 r − 1)(2 − 2) and number 2 of edge-disjoint copies of K4 incident to each vertex.