Leanness Computation: Small Values and Special Graph Classes

Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied unde...

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Bibliographic Details
Published in:Discrete Mathematics and Theoretical Computer Science Vol. 26:2; no. Graph Theory; p. 1
Main Authors: Coudert, David, Coulomb, Samuel, Ducoffe, Guillaume
Format: Journal Article
Language:English
Published: DMTCS 01.01.2024
Discrete Mathematics & Theoretical Computer Science
Subjects:
[info.info-cc]computer science [cs]/computational complexity [cs.cc]
[info.info-ds]computer science [cs]/data structures and algorithms [cs.ds]
Algorithms
Analysis
Computational Complexity
Computer Science
Data Structures and Algorithms
geodetic graphs
graph algorithms
Graph theory
leanness
seth-based lower bounds
France
SETH-based lower bounds
Graph algorithms
Leanness
Geodetic graphs
ISSN:1365-8050, 1462-7264, 1365-8050
Online Access:Get full text
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Summary:Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied under different names, such as "interval thinness" and "fellow traveler property". Graphs with leanness equal to 0, a.k.a. geodetic graphs, also have received special attention in Graph Theory. The practical computation of leanness in real-life complex networks has been studied recently (Mohammed et al., COMPLEX NETWORKS'21). In this paper, we give a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph G is at most some small value ℓ; and computing the leanness on specific graph classes. We obtain improved algorithms in some cases, and time complexity lower bounds under plausible hypotheses.
ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.12544