TY - GEN
AB - We prove three principal results. First we exhibit a drawing of $K_{10}$ in the plane for which there do not exist extensions of the edges to simple closed curves with any two curves intersecting at most twice. Second, we exhibit a drawing of $K_9$ that has an extension of its edges to simple closed curves such that any two curves intersect in at most two points, but no extension to simple closed curves has every two curves intersecting in exactly two points. Third, we show that every h-convex drawing (introduced by Arroyo et al, submitted) has extensions of its edges to simple closed curves such that any two curves intersect in exactly two points. Using this result, we show that} a set of three axioms of simple closed curve extensions characterizes h-convexity.
AU - Arroyo Guevara, Alan M
AU - Richter, Bruce
AU - Sunohara, Matthew
ID - 6313
TI - Extending drawings of complete graphs into arrangements of pseudocircles
ER -
TY - JOUR
AB - The crossing number of a graph G is the least number of crossings over all possible drawings of G. We present a structural characterization of graphs with crossing number one.
AU - Silva, AndrĂ©
AU - Arroyo Guevara, Alan M
AU - Richter, Bruce
AU - Lee, Orlando
ID - 6638
JF - Discrete Mathematics
TI - Graphs with at most one crossing
ER -