WALCOM 2008: WALCOM: Algorithms and Computation pp 132-143

Four-Connected Spanning Subgraphs of Doughnut Graphs

(Extended Abstract)
• Md. Rezaul Karim
• Md. Saidur Rahman
Conference paper

DOI: 10.1007/978-3-540-77891-2_13

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4921)
Cite this paper as:
Karim M.R., Rahman M.S. (2008) Four-Connected Spanning Subgraphs of Doughnut Graphs. In: Nakano S., Rahman M.S. (eds) WALCOM: Algorithms and Computation. WALCOM 2008. Lecture Notes in Computer Science, vol 4921. Springer, Berlin, Heidelberg

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