Detecting Holes and Antiholes in Graphs

In this paper we study the problems of detecting holes and antiholes in general undirected graphs, and we present algorithms for these problems. For an input graph G on n vertices and m edges, our algorithms run in O(n + m2) time and require O(nm) space; we thus provide a solution to the open problem posed by Hayward et al. in [17] asking for an O(n4)-time algorithm for finding holes in arbitrary graphs. The key element of the algorithms is the use of the depth-first-search traversal on appropriate auxiliary graphs in which moving between any two adjacent vertices is equivalent to walking along a P4 (i.e., a chordless path on four vertices) of the input graph or on its complement, respectively. The approach can be generalized so that for a fixed constant k ≥ 5 we obtain an O(nk−1)-time algorithm for the detection of a hole (antihole resp.) on at least k vertices. Additionally, we describe a different approach which allows us to detect antiholes in graphs that do not contain chor...

• Type: Journal
• Year: 2007
• Where: ALGORITHMICA
• Authors: Stavros D. Nikolopoulos, Leonidas Palios
• Keywords: ALGORITHMICA 2007, Algorithms, General Undirected Graphs, Input Graph