Maximal Neighborhood Search and Rigid Interval Graphs

A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic Breadth-First Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the so-called Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several different ways. This allows us obtain several linear time multi-sweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3-sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2-sweep MNS certifying recognition algorithm. Submitted: September 2012 Reviewed: April 2013 Revised: May 2013 Accepted: May 2013 Final: May 2013