Bosch, PaulParra Inza, ErnestoRios Villamar, IsmaelSánchez-Santiesteban, José Luis2026-10-052026-10-052025Bosch, P., Parra Inza, E., Rios Villamar, I., & Sánchez-Santiesteban, J. L. (2025). Total Outer-Independent Domination Number: Bounds and Algorithms. Algorithms, 18(3), 159. https://doi.org/10.3390/a18030159https://hdl.handle.net/11447/11226In graph theory, the study of domination sets has garnered significant interest due to its applications in network design and analysis. Consider a graph G(V, E); a subset of its vertices is a total dominating set (TDS) if, for each x ∈ V(G), there exists an edge in E(G) connecting x to at least one vertex within this subset. If the subgraph induced by the vertices outside the TDS has no edges, the set is called a total outer-independent dominating set (TOIDS). The total outer-independent domination number, denoted as γoit (G), represents the smallest cardinality of such a set. Deciding if a given graph has a TOIDS with at most r vertices is an NP-complete problem. This study introduces new lowerand upper bounds for γoit (G) and presents an exact solution approach using integer linear programming (ILP). Additionally, we develop a heuristic and a procedure to efficiently obtain minimal TOIDS.18 p.enGraph theoryTotal outer-independent dominating setInteger linear programHeuristic algorithmTime complexityArticlehttps://doi.org/10.3390/a18030159