Nonplanar topological inference and political-map graphs

Zhi Zhong Chen; Xin He; Ming Yang Kao

Nonplanar topological inference and political-map graphs

Zhi Zhong Chen^*, Xin He, Ming Yang Kao

^*Corresponding author for this work

Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

17 Scopus citations

Abstract

Given a political map M, we define a mixed graph G where the vertices correspond to the districts on M, the arcs encode the inclusion relation between the districts, and the edges indicate the sharing of boundary points of the districts. We prove that such graphs can be recognized in nondeterministic polynomial time. We then prove that a natural subclass of such graphs can be recognized in O(nα(n) log n) time, where n is the number of vertices in the input graph and α(n) is an inverse of Ackermann's function. The main result is an O(nα(n)log n)-time algorithm for the following problem II: Given a planar graph G = (V, E) and a partition V₁, V₂, ..., V_q of V, decide whether G has a planar embedding ε such that for every V_i, there is a face V_i in ε whose boundary intersects every connected component of the subgraph induced by V_i. This result also has an application in printed circuit board design.

Original language	English (US)
Title of host publication	Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Editors	Anon
Publisher	SIAM
Pages	195-204
Number of pages	10
State	Published - Jan 1 1999
Event	Proceedings of the 1999 10th Annual ACM-SIAM Symposium on Discrete Algorithms - Baltimore, MD, USA Duration: Jan 17 1999 → Jan 19 1999

Other

Other	Proceedings of the 1999 10th Annual ACM-SIAM Symposium on Discrete Algorithms
City	Baltimore, MD, USA
Period	1/17/99 → 1/19/99

ASJC Scopus subject areas

Chemical Health and Safety
Software
Safety, Risk, Reliability and Quality
Discrete Mathematics and Combinatorics

Cite this

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title = "Nonplanar topological inference and political-map graphs",

abstract = "Given a political map M, we define a mixed graph G where the vertices correspond to the districts on M, the arcs encode the inclusion relation between the districts, and the edges indicate the sharing of boundary points of the districts. We prove that such graphs can be recognized in nondeterministic polynomial time. We then prove that a natural subclass of such graphs can be recognized in O(nα(n) log n) time, where n is the number of vertices in the input graph and α(n) is an inverse of Ackermann's function. The main result is an O(nα(n)log n)-time algorithm for the following problem II: Given a planar graph G = (V, E) and a partition V1, V2, ..., Vq of V, decide whether G has a planar embedding ε such that for every Vi, there is a face Vi in ε whose boundary intersects every connected component of the subgraph induced by Vi. This result also has an application in printed circuit board design.",

author = "Chen, {Zhi Zhong} and Xin He and Kao, {Ming Yang}",

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AU - Chen, Zhi Zhong

AU - He, Xin

AU - Kao, Ming Yang

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N2 - Given a political map M, we define a mixed graph G where the vertices correspond to the districts on M, the arcs encode the inclusion relation between the districts, and the edges indicate the sharing of boundary points of the districts. We prove that such graphs can be recognized in nondeterministic polynomial time. We then prove that a natural subclass of such graphs can be recognized in O(nα(n) log n) time, where n is the number of vertices in the input graph and α(n) is an inverse of Ackermann's function. The main result is an O(nα(n)log n)-time algorithm for the following problem II: Given a planar graph G = (V, E) and a partition V1, V2, ..., Vq of V, decide whether G has a planar embedding ε such that for every Vi, there is a face Vi in ε whose boundary intersects every connected component of the subgraph induced by Vi. This result also has an application in printed circuit board design.

AB - Given a political map M, we define a mixed graph G where the vertices correspond to the districts on M, the arcs encode the inclusion relation between the districts, and the edges indicate the sharing of boundary points of the districts. We prove that such graphs can be recognized in nondeterministic polynomial time. We then prove that a natural subclass of such graphs can be recognized in O(nα(n) log n) time, where n is the number of vertices in the input graph and α(n) is an inverse of Ackermann's function. The main result is an O(nα(n)log n)-time algorithm for the following problem II: Given a planar graph G = (V, E) and a partition V1, V2, ..., Vq of V, decide whether G has a planar embedding ε such that for every Vi, there is a face Vi in ε whose boundary intersects every connected component of the subgraph induced by Vi. This result also has an application in printed circuit board design.

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Nonplanar topological inference and political-map graphs

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ASJC Scopus subject areas

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