Recognizing HHD-free and Welsh-Powell Opposition Graphs

In this paper, we consider the recognition problem on two classes of perfectly orderable graphs, namely, 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( min {nmα(n), nm + n2 log n}) time whether a given graph G on n vertices and m edges contains a house or a hole; this leads to an O( min {nmα(n), nm+n2 logn})-time and O(n+m)-space algorithm for recognizing HHD-free graphs. We also show that determining whether the complement $\skew3\overline{G}$ of the graph G contains a house or a hole can be efficiently resolved in O(nm) time using O(n2) space; this in turn leads to an O(nm)-time and O(n2)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HHD-free and WPO-graphs required O(n3) time and O(n2) space.