On the signed total Roman domination and domatic numbers of graphs

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• 作者：Lutz Volkmann ^{volkm@math2.rwth-aachen.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Signed total Roman dominating function ; Signed total Roman domination number ; Signed total Roman domatic number
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：11 December 2016
• 年：2016
• 卷：214
• 期：Complete
• 页码：179-186
• 全文大小：422 K

文摘

A signed total Roman dominating function   on a graph G$G$ is a function f:V(G)⟶{−1,1,2}$f:V\left(G\right)⟶\{-1,1,2\}$ such that ∑_u∈N(v)f(u)≥1${\sum }_{u\in N\left(v\right)}f\left(u\right)\ge 1$ for every vertex v∈V(G)$v\in V\left(G\right)$, where N(v)$N\left(v\right)$ is the neighborhood of v$v$, and every vertex u∈V(G)$u\in V\left(G\right)$ for which f(u)=−1$f\left(u\right)=-1$ is adjacent to at least one vertex w$w$ for which f(w)=2$f\left(w\right)=2$. The signed total Roman domination number   of a graph G$G$, denoted by γ_stR(G)${\gamma }_{stR}\left(G\right)$, equals the minimum weight of a signed total Roman dominating function. A set {f₁,f₂,…,f_d}$\{f_1,f_2,\dots ,f_d\}$ of distinct signed total Roman dominating functions on G$G$ with the property that ${\sum }_{i=1}^d{f}_i\left(v\right)\le 1$ for each v∈V(G)$v\in V\left(G\right)$, is called a signed total Roman dominating family   (of functions) on G$G$. The maximum number of functions in a signed total Roman dominating family on G$G$ is the signed total Roman domatic number   of G$G$, denoted by d_stR(G)$d_{stR}\left(G\right)$. In this paper we continue the investigation of the signed total Roman domination number, and we initiate the study of the signed total Roman domatic number in graphs. We present sharp bounds for γ_stR(G)${\gamma }_{stR}\left(G\right)$ and d_stR(G)$d_{stR}\left(G\right)$. In addition, we determine the total signed Roman domatic number of some graphs.