Interval-Based Graph Representations for Efficient Web Querying

We investigate natural extensions of the Continuous Image Graph (CIG) labelling scheme, which is based on the Consecutive Ones Property of matrices. The CIG labelling scheme generalizes tree labelling schemes to many graphs yet remaining as efficient, i.e., adjacency tests run in constant time, the space requirement is constant per node, and testing whether a graph is a CIG is possible in polynomial time. The CIG labelling scheme is based on finding an ordering of the nodes of a graph such that every node has all its children in one consecutive interval under the ordering. Our extensions cover labelling schemes that are applicable to more graphs yet retain the same efficiency. We propose extensions of the CIG labelling scheme such that • multiple intervals per node, • multiple orderings per graph, • multi-dimensional intervals per node, or • any combination of the above are allowed. We show that our extensions indeed cover more graphs and give (proper) hierarchies of labelling schemes with ever-increasing space needs, covering even more graphs. Unfortunately, however, we also show that each of these natural extensions yields an NPcomplete recognition problem, i.e., testing whether a given labelling scheme of ours is applicable to a given graph is NP-complete. Our results strongly suggest that the CIG labelling scheme is just restricted enough to still be recognizable in polynomial time and that for significantly more general labelling schemes most likely only approximate recognition algorithms are efficient.