Chapter

Computing and Combinatorics

Volume 7434 of the series Lecture Notes in Computer Science pp 335-346

Fáry’s Theorem for 1-Planar Graphs

  • Seok-Hee HongAffiliated withUniversity of Sydney
  • , Peter EadesAffiliated withUniversity of Sydney
  • , Giuseppe LiottaAffiliated withUniversity of Perugia
  • , Sheung-Hung PoonAffiliated withNational Tsing Hua University

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Abstract

A plane graph is a graph embedded in a plane without edge crossings. Fáry’s theorem states that every plane graph can be drawn as a straight-line drawing, preserving the embedding of the plane graph. In this paper, we extend Fáry’s theorem to a class of non-planar graphs. More specifically, we study the problem of drawing 1-plane graphs with straight-line edges. A 1-plane graph is a graph embedded in a plane with at most one crossing per edge. We give a characterisation of those 1-plane graphs that admit a straight-line drawing. The proof of the characterisation consists of a linear time testing algorithm and a drawing algorithm. Further, we show that there are 1-plane graphs for which every straight-line drawing has exponential area. To the best of our knowledge, this is the first result to extend Fáry’s theorem to non-planar graphs.