A vertex incremental approach for dynamically maintaining chordal graphs

Abstract

For a chordal graph G, we study the problem of whether a new vertex u and a given set of edges between u and vertices of G can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, define a maximal subset of the proposed edges that can be added, or conversely a minimal set of extra edges that should be added in addition to the given set. Based on these results, we present a new algorithm which computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O(nm) time. This time complexity matches the best known time bound for minimal triangulation, using a totally new vertex incremental approach. In opposition to previous algorithms, our process adds each new vertex without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added at further steps. © Springer-Verlag Berlin Heidelberg 2003.