Properties of Arrangements Graphs

Article dans revue scientifique avec comité de lecture.An arrangement graph G is the abstract graph obtained from an arrangement of lines L, in general position, by associating vertices of G with the intersection points of L, and the edges of G with the line segments joining the intersection points of L. A simple polygon (respectively path) of n sides in general position, induces a set of n lines by extension of the line segments into lines. The main results of this paper are : (i) given a graph G, it is NP-Hard to determine if G is the arrangement graph of some set of lines (ii) there are non-Hamiltonian arrangement graphs for arrangements of six lines and for odd values of n>6 lines, (iii) all arrangements of n lines contain a subarrangement of size sqrt(n-1)+1 with an inducing polygon (iv) all arrangements on n lines contain an inducing path consisting of n segments and (v) all arrangements on n hyperplanes in R^d contain a simple inducing polygonal cycle of size n