# Graph Searches and Their End Vertices

## TL;DR

This paper provides a comprehensive study of the end vertex problem in various graph search algorithms, offering polynomial and NP-complete results, as well as efficient algorithms for specific graph classes.

## Contribution

It characterizes the end vertex problem for multiple graph search algorithms across different graph classes, including polynomial solutions and NP-completeness proofs.

## Key findings

- Polynomial-time algorithm for maximum cardinality search on chordal graphs
- NP-completeness of the end vertex problem on weakly chordal graphs
- Linear-time algorithms for BFS and LexDFS on specific graph classes

## Abstract

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 $2^n\cdot n^{O(1)}$-time algorithms for deciding the end vertices of breadth-first searches, depth-first searches, maximum cardinality searches, and maximum neighborhood searches on general graphs.