Parallel Algorithms For Recognizing P5-Free And P5-Free Weakly Chordal Graphs

We prove algorithmic characterizations of weakly chordal graphs, which lead to efficient parallel algorithms for recognizing P5-free and ${\bar P}_5$-free weakly chordal graphs. For an input graph on n vertices and m edges, our algorithms run in O(log2n) time and require O(m2/log n) processors on the EREW PRAM model of computation. The proposed recognition algorithms efficiently detect P5s and ${\bar P}_5{\rm s}$ in weakly chordal graphs in O(log n) time with O(m2/logn) processors on the EREW PRAM. Additionally, we show how the algorithms can be augmented to provide a certificate for the existence of a P5 (or a ${\bar P}_5$) in case the input graph is not P5-free (respectively, ${\bar P}_5$-free) weakly chordal.