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Recognizing HH-free, HHD-free, and Welsh-Powell opposition graphs. (English) Zbl 1153.05331

Summary: In this paper, we consider the recognition problem on three classes of perfect graphs, namely, the HH-free, the HHD-free, and the Welsh–Powell opposition graphs (or WPO-graphs). In particular, we prove properties of the chordal completion of a graph and show that a modified version of the classic linear-time algorithm for testing for a perfect elimination ordering can be efficiently used to determine in \(O(n\, {\text{min}}\{m \alpha (n,n), m + n \log n\})\) time whether a given graph \(G\) on \(n\) vertices and \(m\) edges contains a house or a hole; this implies an \(O(n\,{\text{min}}\{m \alpha (n,n), m + n \log n\})\)-time and \(O(n + m)\)-space algorithm for recognizing HH-free graphs, and in turn leads to an HHD-free graph recognition algorithm exhibiting the same time and space complexity. We also show that determining whether the complement \(\overline G\) of the graph \(G\) is HH-free can be efficiently resolved in \(O(nm)\) time using \(O(n^{2})\) space, which leads to an \(O(nm)\)-time and \(O(n^{2})\)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required \(O(n^{3})\) time and \(O(n^{2})\) space.

MSC:

05C85 Graph algorithms (graph-theoretic aspects)
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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