DeepAI AI Chat
Log In Sign Up

Detours in Directed Graphs

01/10/2022
by   Fedor V. Fomin, et al.
0

We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bezáková et al. proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^O (k) n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^O(k) n^O(1) time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path Above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k (diam(G) denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path Above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path Above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path Above Diameter is solvable in polynomial time for each k∈1,…, 4 and is NP-complete for every k≥5. The parameterized complexity of Longest Path Above Diameter on general directed graphs remains an interesting open problem.

READ FULL TEXT

page 1

page 2

page 3

page 4

01/15/2023

Long Directed Detours: Reduction to 2-Disjoint Paths

We study an "above guarantee" version of the Longest Path problem in dir...
11/09/2017

Distances in and Layering of a DAG

The diameter of an undirected unweighted graph G=(V,E) is the maximum va...
04/20/2018

Planar Steiner Orientation is NP-complete

Many applications in graph theory are motivated by routing or flow probl...
02/27/2018

Parameterized Complexity of Diameter

Diameter--the task of computing the length of a longest shortest path---...
03/25/2022

Independent set reconfiguration on directed graphs

Directed Token Sliding asks, given a directed graph and two sets of pair...
12/22/2022

Parameterizing Path Partitions

We study the algorithmic complexity of partitioning the vertex set of a ...
01/06/2023

Simpler and faster algorithms for detours in planar digraphs

In the directed detour problem one is given a digraph G and a pair of ve...