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A vertex incremental approach for maintaining chordality. (English) Zbl 1087.05054
Summary: For a chordal graph \(G=(V,E)\), we study the problem of whether a new vertex \(u \notin V\) and a given set of edges between \(u\) and vertices in \(V\) 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, specify a maximal subset of the proposed edges that can be added along with \(u\), or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge \(uv\), a characterization of the set of edges incident to \(u\) that also must be added to \(G\) along with \(uv\). We propose a data structure that can compute and add each such set in O\((n)\) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O\((nm)\) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is on-line in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently.

05C85 Graph algorithms (graph-theoretic aspects)
68R10 Graph theory (including graph drawing) in computer science
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