Toward the HananiTutte Theorem for Clustered Graphs
Abstract
The weak variant of HananiTutte theorem says that a graph is planar, if it can be drawn in the plane so that every pair of edges cross an even number of times. Moreover, we can turn such a drawing into an embedding without changing the order in which edges leave the vertices. We prove a generalization of the weak HananiTutte theorem that also easily implies the monotone variant of the weak HananiTutte theorem by Pach and Tóth. Thus, our result can be thought of as a common generalization of these two neat results. In other words, we prove the weak HananiTutte theorem for strip clustered graphs, whose clusters are linearly ordered vertical strips in the plane and edges join only vertices in the same cluster or in neighboring clusters with respect to this order. In order to prove our main result we first obtain a forbidden substructure characterization of embedded strip clustered planar graphs. The HananiTutte theorem says that a graph is planar, if it can be drawn in the plane so that every pair of edges not sharing a vertex cross an even number of times. We prove the variant of HananiTutte theorem for strip clustered graphs if the underlying abstract graph is three connected or a tree. In the case of trees our result implies that cplanarity for flat clustered graphs with three clusters is solvable in a polynomial time if the underlying abstract graph is a tree. The proof of the latter result combines our forbidden substructure characterization of embedded strip clustered planar graphs with Tucker's characterization of 01 matrices with consecutive ones property.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.3022
 Bibcode:
 2014arXiv1410.3022F
 Keywords:

 Computer Science  Computational Geometry;
 Mathematics  Combinatorics
 EPrint:
 Section 3 thoroughly revised