Universal Slope Sets for 1-Bend Planar Drawings

نویسندگان

  • Patrizio Angelini
  • Michael A. Bekos
  • Giuseppe Liotta
  • Fabrizio Montecchiani
چکیده

We describe a set of ∆ − 1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree ∆ ≥ 4; this establishes a new upper bound of ∆ − 1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree ∆ has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3 2 (∆ − 1) (the known lower bound being 3 4 (∆ − 1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(∆) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is π (∆−1) .

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A Universal Slope Set for 1-Bend Planar Drawings

We describe a set of ∆−1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree ∆ ≥ 4; this establishes a new upper bound of ∆− 1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree ∆ has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slop...

متن کامل

Universal Slope Sets for Upward Planar Drawings

We prove that every set S of ∆ slopes containing the horizontal slope is universal for 1-bend upward planar drawings of bitonic st-graphs with maximum vertex degree ∆, i.e., every such digraph admits a 1-bend upward planar drawing whose edge segments use only slopes in S. This result is worst-case optimal in terms of the number of slopes, and, for a suitable choice of S, it gives rise to drawin...

متن کامل

Universal Sets of n Points for 1-Bend Drawings of Planar Graphs with n Vertices

This paper shows that any planar graph with n vertices can be point-set embedded with at most one bend per edge on a universal set of n points in the plane. An implication of this result is that any number of planar graphs admit a simultaneous embedding without mapping with at most one bend per edge.

متن کامل

Universal Sets of n Points for One-bend Drawings of Planar Graphs with n Vertices

This paper shows that any planar graph with n vertices can be point-set embedded with at most one bend per edge on a universal set of n points in the plane. An implication of this result is that any number of planar graphs admit a simultaneous embedding without mapping with at most one bend per edge.

متن کامل

On the Total Number of Bends for Planar Octilinear Drawings

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at 45◦ line-segments. For such drawings to be readable, special care is needed in order to keep the number of bends small. As the problem of finding planar octilinear drawings of minimum number of bends is NP-hard [16], in this paper we focus on upper and lower bounds. F...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • CoRR

دوره abs/1703.04283  شماره 

صفحات  -

تاریخ انتشار 2017