American Mathematical Monthly
Algorithmic geometry
Complexes of t-Colorable Graphs
SIAM Journal on Discrete Mathematics
The Polytope of Non-Crossing Graphs on a Planar Point Set
Discrete & Computational Geometry
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Given a convex n-gon P in R^2 with vertices in general position, it is well known that the simplicial complex @q(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n-3. We prove that for any non-convex polygonal region P with n vertices and h+1 boundary components, @q(P) is a ball of dimension n+3h-4. We also provide a new proof that @q(P) is a sphere when P is convex with vertices in general position.