TOTAL ROMAN DOMINATION IN GRAPHS

• Published in: Applicable analysis and discrete mathematics Vol. 10; no. 2; pp. 501 - 517
• Main Authors: Ahangar, Hossein Abdollahzadeh, Henning, Michael A., Samodivkin, Vladimir, Yero, Ismael G.
• Format: Journal Article
• Language: English
• Published: University of Belgrade 01.10.2016
• Subjects: Cardinality; Discrete mathematics; Double stars; Graph theory; Leaves; Literature; Mathematical functions; Mathematical minima; Mathematical theorems; Vertices
• ISSN: 1452-8630; 2406-100X; 2406-100X
• DOI: 10.2298/AADM160802017A

A Roman dominating function on a graph 𝐺 is a function 𝑓: 𝑉 (𝐺) → {0, 1, 2} satisfying the condition that every vertex 𝑢 for which 𝑓(𝑢) = 0 is adjacent to at least one vertex 𝑣 for which 𝑓(𝑣) = 2. The weight of a Roman dominating function 𝑓 is the sum, Σ𝑢∈𝑉 (𝐺)𝑓(𝑢), of the weights of the vertices. The Roman domination number is the minimum weight of a Roman dominating function in 𝐺. A total Roman domination function is a Roman dominating function with the additional property that the subgraph of 𝐺 induced by the set of all vertices of positive weight has no isolated vertex. The total Roman domination number is the minimum weight of a total Roman domination function on 𝐺. We establish lower and upper bounds on the total Roman domination number. We relate the total Roman domination to domination parameters, including the domination number, the total domination number and Roman domination number.