graph visualization developers

Abstract: We study affine arrangements of an even number $n$ of lines and pseudolines with many triangles. For simple ones, Blanc's bound limits the number of triangular faces to $\lfloor n(n-5/2)/3\rfloor$. Adapting the Bartholdi-Blanc-Loisel doubling to the even case, we construct an infinite series of arrangements of $n=6\cdot2^t+2$ ($8,14,26,50,\ldots$) straight lines with two triple points and $(n(n-5/2)+1)/3$ triangles, one more than the simple bound allows. We obtain exhaustive enumerations of triangle-maximal pseudoline arrangements, simple or not, for even $n\le12$. For $n=6,8,10,12,14$ the maximum number of triangles, for both lines and pseudolines, is $7,15,25,38,54$; for $n=8,12,14$ every optimal arrangement has multiple points. All optimal arrangements for even $n\le12$ are realizable by straight lines, except two arrangements of 10 pseudolines, which are not stretchable. For the odd value $n=11$, no arrangement with $33$ triangles exists even with multiple points. We also obtain even arrangements by adding a line to odd ones. For $n\equiv2\pmod 6$ this again gives one triangle more than the simple bound, in pseudolines up to $n=44$ and in straight lines for $n=20$ and $n=32$. Doubling these pseudoline arrangements gives members of the series $18\cdot2^t+2$ and $30\cdot2^t+2$ with the same excess, up to $n=146$ and $n=242$, respectively; those with $n=38$ and $n=74$ are realizable by straight lines. These results lead us to conjecture that an arrangement of an even number $n$ of pseudolines has at most $\lfloor n(n-2)/3\rfloor-\lfloor n/6\rfloor$ triangles.

Abstract: Given a finite graph $G$, we characterize the automorphisms and automorphism groups of $G$ that can be realized as geometric symmetries and symmetry groups of straight line drawings of $G$ in the plane. In doing so, we correct a previous version of this characterization from the literature. As an application, we show that the graphs recently constructed by the first two authors to disprove the Polycirculant Conjecture, which have orders $16{,}464^k$ for each positive integer $k$, are the first known vertex-transitive graphs with no non-trivial realizable geometric symmetries. We also construct infinite families of graphs that exhibit all possible kinds of geometric symmetry.