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arXiv · 2203.04253

Oriented Diameter of Planar Triangulations

Debajyoti Mondal·N. Parthiban·Indra Rajasingh
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Abstract

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 $n/2+O(\sqrt{n})$ upper bound on the oriented diameter of planar triangulations. 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 $k$. In contrast, we consider a weighted version of the oriented diameter problem and show it to be is weakly NP-complete for planar graphs with bounded pathwidth.

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Debajyoti Mondal, N. Parthiban, Indra Rajasingh. 2022-03-08. Oriented Diameter of Planar Triangulations. https://arxiv.org/abs/2203.04253

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