Chapter

WALCOM: Algorithms and Computation

Volume 4921 of the series Lecture Notes in Computer Science pp 132-143

Four-Connected Spanning Subgraphs of Doughnut Graphs

(Extended Abstract)
  • Md. Rezaul KarimAffiliated withDept. of Computer Science and Engineering, Bangladesh University of Engineering and Technology(BUET), Dhaka-1000, Bangladesh. Also Dept. of Computer Science and Engineering, University of Dhaka, Dhaka-1000
  • , Md. Saidur RahmanAffiliated withDept. of Computer Science and Engineering, Bangladesh University of Engineering and Technology(BUET)

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Abstract

The class doughnut graphs is a subclass of 5-connected planar graphs. In a planar embedding of a doughnut graph of n vertices there are two vertex-disjoint faces each having exactly n/4 vertices and each of all the other faces has exactly three vertices. Recently the class of doughnut graphs is introduced to show that a graph in this class admits a straight-line grid drawing with linear area and hence any spanning subgraph of a doughnut graph also admits a straight-line grid drawing with linear area. But recognition of a spanning subgraph of a doughnut graph is a non-trivial problem, since recognition of a spanning subgraph of a given graph is an NP-complete problem in general. In this paper, we establish a necessary and sufficient condition for a 4-connected planar graph G to be a spanning subgraph of a doughnut graph. We also give a linear-time algorithm to augment a 4-connected planar graph G to a doughnut graph if G satisfies the necessary and sufficient condition.

Keywords

Planar Graph Doughnut Graph Straight-Line Drawing Grid Drawing Linear Area Drawing