Abstract
Graph Theory In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic Breadth-First Search (LBFS) and prove its correctness. They believe that their corresponding 5-sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4-sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.
Highlights
In this paper, all graphs are assumed to be finite, simple, undirected and loopless
We recently develop this work of Corneil in [35] by inventing a new variant of Lexicographic Breadth-First Search (LBFS) and a byproduct of this effort is a linear time 2-sweep LBFS certifying algorithm for recognizing unit interval graphs [35, §2]
The LBFS σ-slice in G at time k, which is denoted by Sσ,G(k), or Sσ(k) if the graph G is clear from context, is the set of maximal elements for the partial order
Summary
All graphs are assumed to be finite, simple, undirected and loopless. For any graph G, a map I that assigns to each vertex x ∈ V (G) a nonempty closed interval I(x) = [ I(x), rI(x)] is called an interval representation of G provided xy ∈ E(G) if and only if x = y and I(x) ∩ I(y) = ∅ for all x, y ∈ V (G). Lexicographic Breadth-First Search, often abbreviated as LBFS, is a linear time algorithm for ordering the vertex set of a given graph and is developed by Rose, Tarjan and Lueker in 1976 [5, 13, 46]. Corneil et al [11] suggest a variant of LBFS called LBFS∗ To break tie, it makes use of the information from the previous two vertex orderings. We recently develop this work of Corneil in [35] by inventing a new variant of LBFS and a byproduct of this effort is a linear time 2-sweep LBFS certifying algorithm for recognizing unit interval graphs [35, §2]. We remark that our paper suggests a 3-sweep LBFS algorithm for recognizing interval graphs, though we do not see any linear time implementation of it.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
