A note on locating-dominating sets in twin-free graphs

Nicolas Bousquet, Quentin Chuet, Victor Falgas-Ravry, Amaury Jacques, Laure Morelle

Abstract: In this short note, we prove that every twin-free graph on n vertices contains a locating-dominating set of size at most ⌈58n⌉<math><mo stretchy="false" is="true">⌈</mo><mfrac is="true"><mrow is="true"><mn is="true">5</mn></mrow><mrow is="true"><mn is="true">8</mn></mrow></mfrac><mi is="true">n</mi><mo stretchy="false" is="true">⌉</mo></math>. This improves the earlier bound of ⌊23n⌋<math><mo stretchy="false" is="true">⌊</mo><mfrac is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mn is="true">3</mn></mrow></mfrac><mi is="true">n</mi><mo stretchy="false" is="true">⌋</mo></math> due to Foucaud, Henning, Löwenstein and Sasse from 2016, and makes some progress towards the well-studied locating-dominating conjecture of Garijo, González and Márquez.