### Recognizing HH-free, HHD-free, and Welsh-Powell Opposition Graphs

*Stavros D. Nikolopoulos, Leonidas Palios*

#### Abstract

In this paper, we consider the recognition problem on three classes of perfectly orderable 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 min {m α(n,n), m + n2 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 min {m α(n,n), m + n2 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 G
of the graph G is HH-free can be efficiently resolved in O(n m) time using O(n2) space, which leads to an O(n m)-time and O(n2)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required O(n3) time and O(n2) space.

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