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 + n^2 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 + n^2 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 øverlineG of the graph G is HH-free can be efficiently resolved in O(n m) time using O(n^2) space, which leads to an O(n m)-time and O(n^2)-space algorithm for recognizing WPO-graphs. The previously best algorithms for recognizing HH-free, HHD-free, and WPO-graphs required O(n^3) time and O(n^2) space.
Highlights
A linear order ≺ on the vertices of a graph G is perfect if the ordered graph (G, ≺) contains no induced P4 abcd with a ≺ b and d ≺ c
The interest in perfectly orderable graphs comes from the fact that several problems in graph theory, which are NP-complete in general graphs, have polynomial-time solutions in graphs that admit a perfect order [1; 5]; it is NP-complete to decide whether a graph admits a perfect order [12]
Our HHD-free graph recognition algorithm is motivated by the corresponding algorithm of Hoang and Sritharan [9], which in turn is motivated by the work of Hoang and Khouzam [8] and relies on the following characterization of HHD-free graphs proved by Jamison and Olariu: Theorem 4.1 (Jamison and Olariu [10]) The following two statements are equivalent: (i) The graph G is HHD-free; (ii) For every induced subgraph H of the graph G, every ordering of vertices of H produced by LexBFS is a semi-perfect elimination
Summary
A linear order ≺ on the vertices of a graph G is perfect if the ordered graph (G, ≺) contains no induced P4 abcd with a ≺ b and d ≺ c (such a P4 is called an obstruction). Described recognition algorithms for several classes of perfectly orderable graphs, among which a recognition algorithm for HHP-free graphs; a graph is HHP-free if it contains no hole, no house, and no “P” as induced subgraphs (see Figure 1). The characterization of HHDA-free graphs due to Olariu (a graph G is HHDA-free if and only if every induced subgraph of G either is triangulated or contains a non-trivial module [14]) and the use of modular decomposition [11] allowed Eschen et al to present an O(n m)-time recognition algorithm for HHP-free graphs. For the class of WPO-graphs, Olariu and Randall [15] gave the following characterization: a graph G is WPO-graph if and only if G contains no induced C5 (i.e., a hole on 5 vertices), house, P5, or “P” (see Figure 1) It follows that G is a WPO-graph if and only if G is HHP-free and G is HH-free.
Sign-in/Register to view highlights & summary 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 callMore From: Discrete Mathematics & Theoretical Computer Science
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
