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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.

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