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Outer independent signed double Roman domination
Journal of Applied Mathematics and Computing ( IF 2.4 ) Pub Date : 2021-04-10 , DOI: 10.1007/s12190-021-01535-8
H. Abdollahzadeh Ahangar , F. Nahani Pour , M. Chellali , S. M. Sheikholeslami

Suppose \([3]=\{0,1,2,3\}\) and \([3^{-}]=\{-1,1,2,3\}\). An outer independent signed double Roman dominating function (OISDRDF) of a graph \(\Gamma \) is function \(l:V({\Gamma })\rightarrow [3^{-}]\) for which (i) each vertex t with \(l(t)=-1\) is joined to at least two vertices labeled a 2 or to at least one vertex z with \(l(z)=3\), (ii) each vertex t with \(l(t)=1\) is joined to at least a vertex z with \(l(z)\ge 2,\) (iii) \(l(N[t])=\sum _{w\in N[t]}l(w)\ge 1\) occurs for each vertex t, (iv) the set of vertices labeled \(-1\) under l is an independent set. The weight of an OISDRDF is the sum of its function values over all vertices, and the outer independent signed double Roman domination number (OISDRD-number) \(\gamma _{sdR}^{oi}(\Gamma )\) is the minimum weight of an OISDRDF on \(\Gamma \). We first show that determining the number \(\gamma _{sdR}^{oi}(\Gamma )\) is NP-complete for bipartite and chordal graphs. Then we provide exact values of this parameter for paths and cycles. Moreover, we show that for trees T of order \(n\ge 3,\) \(\gamma _{sdR}^{oi}(\Gamma )\le n-1,\) and we characterize extremal trees attaining this bound.



中文翻译:

外部独立签署双重罗马统治

假设\([3] = \ {0,1,2,3 \} \)\([3 ^ {-}] = \ {-1,1,2,3 \} \)。图\(\ Gamma \)的外部独立有符号双罗曼支配函数(OISDRDF)是函数\(l:V({\ Gamma})\ rightarrow [3 ^ {-}] \),每个(i)顶点\(L(T)= - 1 \)被接合到一个标记为2的至少两个顶点,或者至少一个顶点ž\(升(Z)= 3 \) ,(ⅱ)每个顶点\(l(t)= 1 \)至少与一个具有\(l(z)\ ge 2 \\)(iii)的顶点z连接\ [l(N [t])= \ sum _ {w \对于每个顶点t,在N [t]} l(w)\ ge 1 \)中发生,(iv)在l下标为\(-1 \)的顶点集合是一个独立集合。OISDRDF的权重是其所有顶点上的函数值的总和,外部独立有符号双罗马统治数(OISDRD-number)\(\ gamma _ {sdR} ^ {oi}(\ Gamma)\)\(\ Gamma \)上的OISDRDF的最小权重。我们首先表明,对于二部图和弦图,确定数字\(\ gamma _ {sdR} ^ {oi}(\ Gamma)\)是NP完全的。然后,我们为路径和循环提供此参数的精确值。此外,我们证明对于树T\(n \ ge 3,\)\(\ gamma _ {sdR} ^ {oi}(\ Gamma)\ le n-1,\) 并且我们描绘了达到此界限的极树。

更新日期:2021-04-11
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