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arXiv · 2402.07968

Total Roman {2}-Dominating functions in Graphs

H. Abdollahzadeh Ahangar·M. Chellali·S. M. Sheikholeslami·J. C. Valenzuela-Tripodoro
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Abstract

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).}

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H. Abdollahzadeh Ahangar, M. Chellali, S. M. Sheikholeslami, J. C. Valenzuela-Tripodoro. 2024-02-12. Total Roman {2}-Dominating functions in Graphs. https://doi.org/10.7151/dmgt.2316

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