# The strong convexity spectra of grids

## TL;DR

This paper investigates the convexity properties of oriented graphs, proves the NP-completeness of computing the convexity number, and explicitly determines the strong convexity spectrum for grid graphs.

## Contribution

It establishes NP-completeness for the convexity number problem in bipartite oriented graphs and characterizes the strong convexity spectrum of grid graphs.

## Key findings

- NP-completeness holds even for bipartite graphs with large girth
- Determined the strong convexity spectrum of grid graphs $P_n imes P_m$
- Extended previous results to more general classes of graphs

## Abstract

Let $D$ be a connected oriented graph. A set $S \subseteq V(D)$ is convex in $D$ if, for every pair of vertices $x, y \in S$, the vertex set of every $xy$-geodesic, ($xy$ shortest directed path) and every $yx$-geodesic in $D$ is contained in $S$. The convexity number, ${\rm con}(D)$, of a non-trivial oriented graph, $D$, is the maximum cardinality of a proper convex set of $D$. The strong convexity spectrum of the graph $G$, $S_{SC} (G)$, is the set $\{{ \rm con}(D) \colon\ D {\rm \ is \ a \ strong \ orientation \ of \ } G \}$. In this paper we prove that the problem of determining the convexity number of an oriented graph is $\mathcal{NP}$-complete, even for bipartite oriented graphs of arbitrary large girth, extending previous known results for graphs. We also determine $S_{SC} (P_n \Box P_m)$, for every pair of integers $n,m \ge 2$.