Algorithms and complexity for geodetic sets on partial grids.

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in a shortest path between some pair of vertices of S. The MINIMUM GEODETIC SET (MGS) problem is to find a geodetic set with minimum cardinality of a given graph. A grid embedding of a graph is a set of points in two dimensions with integer coordinates such that each point in the set represents a vertex of the graph and, for each edge, the points corresponding to its endpoints are at Euclidean distance 1. A graph is a partial grid if it has a grid embedding. In this paper, we first prove that MINIMUM GEODETIC SET remains NP-hard even for subcubic partial grids of arbitrary girth. This jointly strengthens three existing hardness results: for bipartite graphs (Dourado et al. 2010 [11]), subcubic graphs (Bueno et al. 2018 [4]), and planar graphs (Chakraborty et al. 2020 [6]). The area of an internal face is the number of integer points lying on the boundary or interior of the face. A graph is a solid grid if it has a grid embedding such that all interior faces have area exactly four. To complement the above hardness result, we design a linear-time algorithm for MINIMUM GEODETIC SET on solid grids, improving on a 3-approximation algorithm by Chakraborty et al. (2020) [6].Our results hold for EDGE GEODETIC SET as well. A set S of vertices of a graph G is a geodetic set if every edge of G lies in a shortest path between some pair of vertices of S. The MINIMUM EDGE GEODETIC SET (MEGS) problem is to find an edge geodetic set with minimum cardinality of a given graph. As corollaries, we obtain that MEGS remains NP-hard on partial grids and is linear-time solvable on solid grids.