2025-06-102025-06-102025Valenzuela-Tripodoro, J.C., Mateos-Camacho, M.A., Cera López, M. y Álvarez-Ruiz, M.P. (2025). On the Total Version of Triple Roman Domination in Graphs. MATHEMATICS, 13 (8). https://doi.org/10.3390/math130812772227-7390https://hdl.handle.net/11441/174167In this paper, we describe the study of total triple Roman domination. Total triple Roman domination is an assignment of labels from {0, 1, 2, 3, 4} to the vertices of a graph such that every vertex is protected by at least three units either on itself or its neighbors while ensuring that none of its neighbors remains unprotected. Formally, a total triple Roman dominating function is a function f : V(G) → {0, 1, 2, 3, 4} such that f (N[v]) ≥ |AN(v)| + 3, where AN(v) denotes the set of active neighbors of vertex v, i.e., those assigned a positive label. We investigate the algorithmic complexity of the associated decision problem, establish sharp bounds regarding graph structural parameters, and obtain the exact values for several graph families.application/pdfengAttribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/Roman dominationtotal Roman dominationtriple Roman dominationtotal triple Roman dominationinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccess10.3390/math13081277