On the oriented diameter of planar triangulations

• Published in: Journal of combinatorial optimization Vol. 47; no. 5
• Main Authors: Mondal, Debajyoti, Parthiban, N., Rajasingh, Indra
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2024; Springer Nature B.V
• Subjects: Apexes; Combinatorics; Convex and Discrete Geometry; Diameters; Graph theory; Graphs; Lower bounds; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Shortest-path problems; Theory of Computation; Upper bounds; Triangular grid; Oriented diameter; Planar graph; Separator
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-024-01177-z

The diameter of an undirected or a directed graph is defined to be the maximum shortest path distance over all pairs of vertices in the graph. Given an undirected graph G , we examine the problem of assigning directions to each edge of G such that the diameter of the resulting oriented graph is minimized. The minimum diameter over all strongly connected orientations is called the oriented diameter of G . The problem of determining the oriented diameter of a graph is known to be NP-hard, but the time-complexity question is open for planar graphs. In this paper we compute the exact value of the oriented diameter for triangular grid graphs. We then prove an n /3 lower bound and an n / 2 + O n upper bound on the oriented diameter of planar triangulations, where n is the number of vertices in G . It is known that given a planar graph G with bounded treewidth and a fixed positive integer k , one can determine in linear time whether the oriented diameter of G is at most k . We consider a weighted version of the oriented diameter problem and show it to be weakly NP-complete for planar graphs with bounded pathwidth.