Compact visibility representation of 4-connected plane graphs

  • Authors:

  • Xin He;Jiun-Jie Wang;Huaming Zhang

  • Affiliations:

  • Department of Computer Science and Engineering, University at Buffalo, Buffalo, NY;Department of Computer Science and Engineering, University at Buffalo, Buffalo, NY;Department of Computer Science, University of Alabama in Huntsville, Huntsville, AL

  • Venue:
  • COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part I
  • Year:
  • 2010

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Abstract

The visibility representation (VR for short) is a classical representation of plane graphs. The VR has various applications and has been extensively studied. A main focus of the study is to minimize the size of the VR. It is known that there exists a plane graph G with n vertices where any VR of G requires a size at least ⌊2n/3⌋ × (⌊4n/3⌋-3). For upper bounds, it is known that every plane graph has a VR with size at most ⌊2/3n⌋ × (2n - 5), and a VR with size at most (n - 1) × ⌊4/3n⌋. It has been an open problem to find a VR with both height and width simultaneously bounded away from the trivial upper bounds (namely of size chn×cwn with ch cw n/4 + 2⌈√n⌉ + 4 and width ≤ ⌈3n/2⌉. Our VR algorithm is based on an st-orientation of 4-connected plane graphs with special properties. Since the st-orientation is a very useful concept in other applications, this result may be of independent interests.