We prove that every planar graph with maximum degree three has a planar drawing in which the edges are drawn as circular arcs that meet at equal angles around every vertex. Our construction is based on the Koebe–Thurston–Andreev circle packing theorem, and uses a novel type of Voronoi diagram for circle packings that is invariant under Möbius transformations, defined using three-dimensional hyp...

Planar graphs can be represented as the intersection of different types geometric objects in plane, e.g., circles (Koebe, 1936), line segments (Chalopin & Gonçalves, SODA 2009), L-shapes (Gonçalves et al., 2018). For general graphs, however, even deciding whether such representations exist is often NP-hard. We consider apex i.e., that made planar by removing one vertex from them. show, somewhat...

Colin de Verdi ere introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G) 2 if and only if G is outerplanar, and (G) 3 if and only if G is planar. We prove that if the complement of a graph G on n nodes is outerplanar, then (G) n ? 4, and if it is planar, then (G) n ? 5. We give a full characterization of maximal planar graphs with (G) = n ? 5. In the opposite di...

Abstract A semigroup generated by two dimensional C contracting maps is considered. We call a such semigroup regular if the maximum K of the conformal dilatations of generators, the maximum l of the norms of the derivatives of generators and the smoothness α of the generators satisfy a compatibility condition K < 1/l. We prove that the shape of the image of the core of a ball under any element ...

A mesh M with planar faces is called an edge offset (EO) mesh if there exists a combinatorially equivalent mesh M d such that corresponding edges of M and M d lie on parallel lines of constant distance d. The edges emanating from a vertex of M lie on a right circular cone. Viewing M as set of these vertex cones, we show that the image of M under any Laguerre transformation is again an EO mesh. ...