Planar Turán numbers of three configurations

The planar Tuán number of H, denoted by ex₏ (n, H), is defined as the maximum number of edges in an n-vertex H-free planar graph. The exact value of ex₏ (n, H) remains a mystery when H is large (for example, H is a long path or a long cycle), while tight bounds have been established for many small planar graphs such as cycles, paths, Θ-graphs and other small graphs formed by a union of them. One representative graph among such union graphs is K₁+L where L is a linear forest without isolated vertices. Previous works solved the cases when L is a path or a matching. In this work, we first investigate the planar Turán number of the graph K₁+L when L is the disjoint union of a P₂ and P₃. Equivalently, K₁+L represents a specific configuration formed by combining a C₃ and a Θ₄. We further consider the planar Turán numbers of the all graphs obtained by combining C₃ and Θ₄. Among the six possible such configurations, three have been resolved in earlier works. For the remaining three configurations (including K₁+ (P₂̇P₃) ), we derive tight bounds. Furthermore, we completely characterize all extremal graphs for the remaining two of these three cases.