Extremal $k$-forcing sets in oriented graphs

TL;DR

This paper introduces and studies the maximum and minimum oriented $k$-forcing numbers in graphs, generalizing existing concepts and establishing relationships with known graph invariants such as the path covering and independence numbers.

Contribution

It defines new extremal oriented $k$-forcing numbers for graphs and relates them to classical graph invariants, extending the theory of $k$-forcing to oriented graphs.

Findings

Minimum oriented $k$-forcing number equals the path covering number.

Maximum oriented $k$-forcing number is at least the independence number.

Results hold with equality for trees or when $k$ exceeds maximum degree.

Abstract

This article studies the \emph{$k$-forcing number} for oriented graphs, generalizing both the \emph{zero forcing number} for directed graphs and the $k$-forcing number for simple graphs. In particular, given a simple graph $G$, we introduce the maximum (minimum) oriented $k$-forcing number, denoted $\MOF_k(G)$ ($\mof_k(G)$), which is the largest (smallest) $k$-forcing number among all possible orientations of $G$. These new ideas are compared to known graph invariants and it is shown that, among other things, $\mof(G)$ equals the path covering number of $G$ while $\MOF_k(G)$ is greater than or equal to the independence number of $G$ -- with equality holding if $G$ is a tree or if $k$ is at least the maximum degree of $G$. Along the way, we also show that many recent results about $k$-forcing number can be modified for oriented graphs.