Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer

نویسندگان

  • Ching-Chi Lin
  • Hsueh-I Lu
  • I-Fan Sun
چکیده

Let G be an n-node planar graph. In a visibility representation of G, each node of G is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained from Schnyder’s realizer for the triangulated G yields a visibility representation of G no wider than ⌊ 22n−40 15 ⌋ . Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant’s open question about whether ⌊ 3n−6 2 ⌋ is a worst-case lower bound on the required width. Also, if G has no degree-three (respectively, degreefive) internal node, then our visibility representation for G is no wider than ⌊ 4n−9 3 ⌋ (respectively, ⌊ 4n−7 3 ⌋ ). Moreover, if G is four-connected, then our visibility representation for G is no wider than n − 1, matching the best known result of Kant and He. As a by-product, we give a much simpler proof for a corollary of Wagner’s theorem on realizers due to Bonichon, Le Saëc, and Mosbah.

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

2 J ul 2 00 1 Orderly Spanning Trees with Applications ∗ Hsueh -

We introduce and study the orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of a plane graph H of G, and an orderly spanning tree of H. We also...

متن کامل

Graphs admitting d-realizers: spanning-tree-decompositions and box-representations

A d-realizer is a collection R = {π1, . . . , πd} of d permutations of a set V representing an antichain in R. We use R to define a graph GR on the suspended set V + = V ∪ {s1, . . . , sd}. It turns out that GR has dn+ ( d 2 ) edges (n = |V |), among them the edges of the outer clique on {s1, . . . , sd}. The inner edges of GR can be partitioned into d trees such that Ti spans V + si. In the ca...

متن کامل

Orderly Spanning Trees with Applications

We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning ...

متن کامل

Graphs admitting d-realizers: tree-decompositions and box-representations

A d-realizer is a collection R = {π1, . . . , πd} of d permutations of a set V representing an antichain in R. We use R to define a graph GR on the suspended set V + = V ∪ {s1, . . . , sd}. It turns out that GR has dn + ( d 2 ) edges (n = |V |), among them the edges of the outer clique on {s1, . . . , sd}. The inner edges of GR can be partitioned into d trees such that Ti spans V + si. In the c...

متن کامل

A More Compact Visibility Representation

16 4-block tree in linear time. The problem is already linear time solvable in case of 2-and 3-connected components (see e.g., Hopcroft & Tarjan 8]), hence solving this open problem yields a nice generalization. As a last subject we consider the method of triangulating planar graphs. It would be interesting to triangulate G such that it is 4-connected. Indeed, G may have separating triangles, i...

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:
  • SIAM J. Discrete Math.

دوره 18  شماره 

صفحات  -

تاریخ انتشار 2003