Eternal dominating sets on digraphs and orientations of graphs

TL;DR

This paper extends the concept of eternal dominating sets to directed graphs, exploring their properties, computational complexity, and specific cases across various graph classes.

Contribution

It generalizes known graph results to digraphs, proves NP-hardness of computing oriented eternal domination, and characterizes graphs with certain domination parameters.

Findings

Computing the oriented eternal dominating number is NP-hard.

Characterization of graphs with oriented m-eternal domination number 2.

Analysis of these parameters on various graph classes such as trees, cycles, and grids.

Abstract

We study the eternal dominating number and the m-eternal dominating number on digraphs. We generalize known results on graphs to digraphs. We also consider the problem "oriented (m-)eternal domination", consisting in finding an orientation of a graph that minimizes its eternal dominating number. We prove that computing the oriented eternal dominating number is NP-hard and characterize the graphs for which the oriented m-eternal dominating number is 2. We also study these two parameters on trees, cycles, complete graphs, complete bipartite graphs, trivially perfect graphs and different kinds of grids and products of graphs.