The $n/2$-bound for locating-dominating sets in subcubic graphs

TL;DR

This paper proves that the location-domination number in subcubic graphs is at most half of the number of vertices, confirming a conjecture and identifying specific exceptions.

Contribution

It establishes the $n/2$-bound for subcubic graphs and extends the result to graphs with certain types of twins, clarifying the bounds for various classes of cubic graphs.

Findings

The $n/2$-bound holds for subcubic graphs, and is tight.

The bound applies to subcubic graphs with open twins of degree 3 and closed twins of any degree.

The bound does not hold for subcubic graphs with open twins of degree 1 or 2.

Abstract

The location-domination number is conjectured to be at most half of the order for twin-free graphs with no isolated vertices. We prove that this conjecture holds and is tight for subcubic graphs. We also show that the same upper bound holds for subcubic graphs with open twins of degree 3 and closed twins of any degree, but not for subcubic graphs with open twins of degree 1 or 2. These results then imply that the same upper bound holds for all cubic graphs (with or without twins) except $K_4$ and $K_{3,3}$.