2 research outputs found
Graph Searches and Their End Vertices
Graph search, the process of visiting vertices in a graph in a specific
order, has demonstrated magical powers in many important algorithms. But a
systematic study was only initiated by Corneil et al.~a decade ago, and only by
then we started to realize how little we understand it. Even the apparently
na\"{i}ve question "which vertex can be the last visited by a graph search
algorithm," known as the end vertex problem, turns out to be quite elusive. We
give a full picture of all maximum cardinality searches on chordal graphs,
which implies a polynomial-time algorithm for the end vertex problem of maximum
cardinality search. It is complemented by a proof of NP-completeness of the
same problem on weakly chordal graphs.
We also show linear-time algorithms for deciding end vertices of
breadth-first searches on interval graphs, and end vertices of lexicographic
depth-first searches on chordal graphs. Finally, we present -time algorithms for deciding the end vertices of breadth-first
searches, depth-first searches, maximum cardinality searches, and maximum
neighborhood searches on general graphs
A Polynomial-Time Algorithm for MCS Partial Search Order on Chordal Graphs
We study the partial search order problem (PSOP) proposed recently by
Scheffler [WG 2022]. Given a graph together with a partial order over the
vertices of , this problem determines if there is an -ordering
that is consistent with the given partial order, where is a graph
search paradigm like BFS, DFS, etc. This problem naturally generalizes the
end-vertex problem which has received much attention over the past few years.
It also generalizes the so-called -tree recognition problem
which has just been studied in the literature recently. Our main contribution
is a polynomial-time dynamic programming algorithm for the PSOP on chordal
graphs with respect to the maximum cardinality search (MCS). This resolves one
of the most intriguing open questions left in the work of Sheffler [WG 2022].
To obtain our result, we propose the notion of layer structure and study
numerous related structural properties which might be of independent interest.Comment: 12 page