Algorithmic complexity of outer independent Roman domination and outer independent total Roman domination

Poureidi, Abolfazl; Ghaznavi, Mehrdad; Fathali, Jafar

doi:10.1007/s10878-020-00682-1

Cited by 2 publications

(2 citation statements)

References 9 publications

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“…After the two papers [9,10] were published, the topic attracted many researchers. Poureidi et al [11] proposed an algorithm to compute γ oiR (G) in O(|V|) time. Martínez et al [12] obtained some bounds on γ oiR (G) in terms of other parameters.…”

Section: Introductionmentioning

confidence: 99%

On Two Outer Independent Roman Domination Related Parameters in Torus Graphs

et al. 2022

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In a graph G=(V,E), where every vertex is assigned 0, 1 or 2, f is an assignment such that every vertex assigned 0 has at least one neighbor assigned 2 and all vertices labeled by 0 are independent, then f is called an outer independent Roman dominating function (OIRDF). The domination is strengthened if every vertex is assigned 0, 1, 2 or 3, f is such an assignment that each vertex assigned 0 has at least two neighbors assigned 2 or one neighbor assigned 3, each vertex assigned 1 has at least one neighbor assigned 2 or 3, and all vertices labeled by 0 are independent, then f is called an outer independent double Roman dominating function (OIDRDF). The weight of an (OIDRDF) OIRDF f is the sum of f(v) for all v∈V. The outer independent (double) Roman domination number (γoidR(G)) γoiR(G) is the minimum weight taken over all (OIDRDFs) OIRDFs of G. In this article, we investigate these two parameters γoiR(G) and γoidR(G) of regular graphs and present lower bounds on them. We improve the lower bound on γoiR(G) for a regular graph presented by Ahangar et al. (2017). Furthermore, we present upper bounds on γoiR(G) and γoidR(G) for torus graphs. Furthermore, we determine the exact values of γoiR(C3□Cn) and γoiR(Cm□Cn) for m≡0(mod4) and n≡0(mod4), and the exact value of γoidR(C3□Cn). By our result, γoidR(Cm□Cn)≤5mn/4 which verifies the open question is correct for Cm□Cn that was presented by Ahangar et al. (2020).

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Section: Introductionmentioning

confidence: 99%

On Two Outer Independent Roman Domination Related Parameters in Torus Graphs

et al. 2022

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“…Independent Roman domination was defined in [9] but Abadi et al were the first to study this topic. Independent (Roman) domination has been studied by several authors [3,[19][20][21][22].…”

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Independent Roman bondage of graphs

et al. 2023

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An independent Roman dominating function (IRD-function) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the conditions that (i) every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$, and (ii) the set of all vertices assigned non-zero values under $f$ is independent. The weight of an IRD-function is the sum of its function values over all vertices, and the independent Roman domination number $i_{R}(G)$ of $G$ is the minimum weight of an IRD-function on $G$. In this paper, we initiate the study of the independent Roman bondage number $b_{iR}(G)$ of a graph $G$ having at least one component of order at least three, defined as the smallest size of set of edges $F\subseteq E(G)$ for which $i_{R}(G-F)>i_{R}(G)$. We begin by showing that the decision problem associated with the independent Roman bondage problem is NP-hard for bipartite graphs. Then various upper bounds on $b_{iR}(G)$ are established as well as exact values on it for some special graphs. In particular, for trees $T$ of order at least three, it is shown that $b_{iR}(T)\leq3,$ while for connected planar graphs the upper bounds are in terms of the maximum degree with refinements depending on the girth of the graph.

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Algorithmic complexity of outer independent Roman domination and outer independent total Roman domination

Cited by 2 publications

References 9 publications

On Two Outer Independent Roman Domination Related Parameters in Torus Graphs

On Two Outer Independent Roman Domination Related Parameters in Torus Graphs

Independent Roman bondage of graphs

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