Total Roman {2}-Dominating functions in Graphs

• Published in: arXiv.org
• Main Authors: H Abdollahzadeh Ahangar, Chellali, M, Sheikholeslami, S M, Valenzuela-Tripodoro, J C
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 12.02.2024
• Subjects: Apexes; Graph theory; Graphs; Mathematical functions; Mathematics - Combinatorics; Minimum weight
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2402.07968

A Roman \(\{2\}\)-dominating function (R2F) is a function \(f:V\rightarrow \{0,1,2\}\) with the property that for every vertex \(v\in V\) with \(f(v)=0\) there is a neighbor \(u\) of \(v\) with \(f(u)=2\), or there are two neighbors \(x,y\) of \(v\) with \(f(x)=f(y)=1\). A total Roman \(\{2\}\)-dominating function (TR2DF) is an R2F \(f\) such that the set of vertices with \(f(v)>0\) induce a subgraph with no isolated vertices. The weight of a TR2DF is the sum of its function values over all vertices, and the minimum weight of a TR2DF of \(G\) is the total Roman \(\{2\}\)-domination number \(\gamma_{tR2}(G).\) In this paper, we initiate the study of total Roman \(\{2\}\)-dominating functions, where properties are established. Moreover, we present various bounds on the total Roman \(\{2\}\)-domination number. We also show that the decision problem associated with \(\gamma_{tR2}(G)\) is NP-complete for bipartite and chordal graphs. {Moreover, we show that it is possible to compute this parameter in linear time for bounded clique-width graphs (including tres).}