Abstract
A k-chordalisation of a graph G = (V,E) is a graph H = (V,F) obtained by adding edges to G, such that H is a chordal graph with maximum clique size at most k. This note considers the problem: given a graph G = (V,E) which pairs of vertices, non-adjacent in G, will be an edge in every k-chordalisation of G. Such a pair is called necessary for treewidth k. An equivalent formulation is: which edges can one add to a graph G such that every tree decomposition of G of width at most k is also a tree decomposition of the resulting graph G′. Some sufficient, and some necessary and sufficient conditions are given for pairs of vertices to be necessary for treewidth k. For a fixed k, one can find in linear time for a given graph G the set of all necessary pairs for treewidth k. If k is given as part of the input, then this problem is coNP-hard. A few similar results are given when interval graphs (and hence pathwidth) are used instead of chordal graphs and treewidth.
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Bodlaender, H.L. Necessary Edges in k-Chordalisations of Graphs. Journal of Combinatorial Optimization 7, 283–290 (2003). https://doi.org/10.1023/A:1027320705349
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DOI: https://doi.org/10.1023/A:1027320705349