On some flag-transitive extensions of the dual petersen graph

We consider flag-transitive extensions of the dual Petersen graph satisfying one of the following “pathological” properties: (¬LL) there are pairs of collinear points incident to more than one line; (¬T) there are triples of collinear points not incident to the same plane. We prove that there is just one example satisfying (¬LL), with six points and S6 as its automorphism group, and there are just four examples satisfying (¬T), with 18, 16, 32, and 32 points and groups 3 - S6, 24 - S5, 25 - S5, and 25 - A5, respectively. The paper is organized as follows. In Section 1 we recall some known results and we explain some motivations of our investigation. The examples we will characterize are described in Section 2. The characterization theorems are stated in Section 3 and proved in Sections 5 and 6. In Section 4 we state some lemmas, to be used in Sections 5 and 6.