Linearizing Partial Search Orders

TL;DR

This paper studies the problem of finding graph search orderings that extend a given partial order, providing polynomial-time algorithms for specific graph classes and search methods, thus advancing understanding of search order construction.

Contribution

It introduces a generalized problem of linearizing partial search orders and offers the first polynomial algorithms for certain search problems on specific graph classes.

Findings

Polynomial-time algorithms for LBFS on chordal bipartite graphs.

Polynomial-time algorithms for LBFS and MCS on split graphs.

First polynomial algorithms for end-vertex and search tree problems in these contexts.

Abstract

In recent years, questions about the construction of special orderings of a given graph search were studied by several authors. On the one hand, the so called end-vertex problem introduced by Corneil et al. in 2010 asks for search orderings ending in a special vertex. On the other hand, the problem of finding orderings that induce a given search tree was introduced already in the 1980s by Hagerup and received new attention most recently by Beisegel et al. Here, we introduce a generalization of some of these problems by studying the question whether there is a search ordering that is a linear extension of a given partial order on a graph's vertex set. We show that this problem can be solved in polynomial time on chordal bipartite graphs for LBFS, which also implies the first polynomial-time algorithms for the end-vertex problem and two search tree problems for this combination of graph class and search. Furthermore, we present polynomial-time algorithms for LBFS and MCS on split graphs which generalize known results for the end-vertex and search tree problems.