A Linear-Time Algorithm for Finding Induced Planar Subgraphs

Abstract

In this paper we study the problem of efficiently and effectively extracting induced planar subgraphs. Edwards and Farr proposed an algorithm with O(mn) time complexity to find an induced planar subgraph of at least 3n/(d+1) vertices in a graph of maximum degree d. They also proposed an alternative algorithm with O(mn) time complexity to find an induced planar subgraph graph of at least 3n/(bar{d}+1) vertices, where bar{d} is the average degree of the graph. These two methods appear to be best known when d and bar{d} are small. Unfortunately, they sacrifice accuracy for lower time complexity by using indirect indicators of planarity. A limitation of those approaches is that the algorithms do not implicitly test for planarity, and the additional costs of this test can be significant in large graphs. In contrast, we propose a linear-time algorithm that finds an induced planar subgraph of n-nu vertices in a graph of n vertices, where nu denotes the total number of vertices shared by the detected Kuratowski subdivisions. An added benefit of our approach is that we are able to detect when a graph is planar, and terminate the reduction. The resulting planar subgraphs also do not have any rigid constraints on the maximum degree of the induced subgraph. The experiment results show that our method achieves better performance than current methods on graphs with small skewness