Independent Roman Research Articles

Trooping the (School) Colour

Trooping the (School) Colour

Independent Roman bondage of graphs

An independent Roman dominating function (IRD-function) on a graph G is a function f : V(G) → {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 iR(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 biR(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 ⊆ E(G) for which iR(G − F) > iR(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 biR(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 biR(T) ≤ 3, while for connected planar graphs the upper bounds are in terms of the maximum degree with refinements depending on the girth of the graph.

Lower and upper bounds on independent double Roman domination in trees

For a graph G = ( V , E ) , a double Roman dominating function (DRDF) f : V → {0, 1, 2, 3} has the property that for every vertex v ∈ V with f ( v )=0 , either there exists a neighbor u ∈ N ( v ) , with f ( u )=3 , or at least two neighbors x , y ∈ N ( v ) having f ( x )= f ( y )=2 , and every vertex with value 1 under f has at least a neighbor with value 2 or 3 . The weight of a DRDF is the sum f ( V )=∑ v ∈ V f ( v ) . A DRDF f is an independent double Roman dominating function (IDRDF) if the vertices with weight at least two form an independent set. The independent double Roman domination number i d R ( G ) is the minimum weight of an IDRDF on G . In this paper, we show that for every tree T with diameter at least three, i ( T )+ i R ( T )−( s ( T ))/2 + 1 ≤ i d R ( T )≤ i ( T )+ i R ( T )+ s ( T )−2 , where i ( T ), i R ( T ) and s ( T ) are the independent domination number, the independent Roman domination number and the number of support vertex of T , respectively.

On Two Outer Independent Roman Domination Related Parameters in Torus Graphs

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).

Independent roman ${3}$-domination

Let $G$ be a simple, undirected graph. In this paper, we initiate the study of independent Roman ${3}$-domination. A function $g : V(G) rightarrow lbrace 0, 1, 2, 3 rbrace$ having the property that $sum_{v in N_G(u)}^{} g(v) geq 3$, if $g(u) = 0$, and $sum_{v in N_G(u)}^{} g(v) geq 2$, if $g(u) = 1$ for any vertex $u in V(G)$, where $N_G(u)$ is the set of vertices adjacent to $u$ in $G$, and no two vertices assigned positive values are adjacent is called an textit{ independent Roman ${3}$-dominating function} (IR3DF) of $G$. The weight of an IR3DF $g$ is the sum $g(V) = sum_{v in V}g(v)$. Given a graph $G$ and a positive integer $k$, the independent Roman ${3}$-domination problem (IR3DP) is to check whether $G$ has an IR3DF of weight at most $k$. We investigate the complexity of IR3DP in bipartite and chordal graphs. The minimum independent Roman $lbrace 3 rbrace$-domination problem (MIR3DP) is to find an IR3DF of minimum weight in the input graph. We show that MIR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs. We also show that the domination problem and IR3DP are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MIR3DP.

Undoing the Knots: Five Generations of American Catholic Anti-Blackness by Maureen H. O’Connell

Reviewed by: Undoing the Knots: Five Generations of American Catholic Anti-Blackness by Maureen H. O’Connell Alex Mikulich II In Undoing the Knots, Maureen O'Connell thoughtfully probes the entanglements of Roman Catholic Anti-Blackness. Yet in her efforts, she fails to address one of its enduring dimensions, namely: coloniality, the hidden, oppressive side of European modernity. Being a colonized and raced person is, by definition, “to be incapable of giving” explains Jane Anna Gordon in her celebration of W.E.B. Du Bois’s “Gift of Double-Consciousness.”1 The colonial epistemology that the Roman Catholic papacy initiated in the early fifteenth century inaugurated both the sinful idolatry of racial superiority and Eurocentric people’s refusal to celebrate the divine giftedness of Black, Brown, and Indigenous peoples. The interconnected economic, ecological, and racial crises of our time remain unabated, in part, because we white settler Americans consume a “happy story” about economic growthism in our socialization systems even as this so-called “free market” impoverishes billions of people and destroys the planet. We white American settlers conveniently forget that the economic rise of the West and the United States depended upon enslavement of African and American Indigenous peoples, expropriating both labor and capital, and dispossessing them of lands and abundant resources. This is the Colonial Matrix Power, aptly named by the sociologist Anibal Quijano, that perpetuates police and vigilante violence against Black people at rates nearly the same as that of lynching just a century ago. This so-called “modern” system colonizes Black, Latino/a, and Indigenous space and bodies through the school to prison pipeline, hyper-incarceration, and by fostering a political economy of segregation through a combination of racially structured banking, real estate, and [End Page 64] governmental policies that “hastens physical decline in urban neighborhoods [and] forever incentivizes their perpetuation.”2 Before any white theologian articulated racial mercy, W.E.B. Du Bois’s gift of double consciousness is only one example of many gifts African peoples offer to practice the porousness and mutuality O’Connell invites (168). Black, Brown, and Indigenous peoples offer love within the binds of oppression, like the gift of double consciousness, inviting white folks to perceive and lament our own complicity in anti-Blackness, and celebrate the divine giftedness, beauty, grandeur, and dignity of Black peoples. A sense of awe before the Creator and celebration of the giftedness of Black people fosters the humility to open ourselves in vulnerability and acknowledge our need for Black people in the intimate, interdependent web of relations of human and non-human kin. Then, perhaps, we might loosen knots of coloniality and begin to sew a new cloth of mutuality and togetherness. When and how will we white folks celebrate the divine giftedness of Black, Brown, and Indigenous peoples and engage the impasse of anti-Black coloniality? Alex Mikulich Independent Roman Catholic social ethicist Footnotes 1. Anna Jane Gordon, “The Gift of Double Consciousness: Some Obstacles to Grasping the Contributions of the Colonized,” in Postcolonialism and Political Theory, ed. Nalini Persram (Lanham, MD: Lexington Books, 2007), 143–163. 2. Keeanga-Yamahtta Taylor, Race for Profit: How Banks and the Real Estate Industry Undermined Black Home Ownership (Chapel Hill: University of North Carolina Press, 2019), 11. Copyright © 2022 American Catholic Historical Society

Further results on the independent Roman domination number of graphs

Let f : V (G) → {0, 1, 2} be a function on a graph G with vertex set V (G). Let Vi = {v ∈ V (G) : f (v) = i} for every i ∈ {0, 1, 2}. The function f is said to be an independent Roman dominating function on G if V 1 ∪ V 2 is an independent set and for every υ 2 V 0. The minimum weight among all independent Roman dominating functions f on G is the independent Roman domination number of G, and is denoted by iR (G). In this paper we continue with the study of this parameter. In particular, we provide new bounds on iR (G) in terms of other domination invariants. Some of our results are tight bounds that improve some well-known results. Finally, we compute the independent Roman domination number of some product graphs, and we provide an alternative proof to show that the problem of computing iR (G) is NP-hard.

Efficient algorithms for independent Roman domination on some classes of graphs

Let $G=(V,E)$ be a given graph of order $n $. A function $f : V to {0,1, 2}$ is an independent Roman dominating function (IRDF) on $G$ if for every vertex $vin V$ with $f(v)=0$ there is a vertex $u$ adjacent to $v$ with $f(u)=2$ and ${vin V:f(v)> 0}$ is an independent set. The weight of an IRDF $f$ on $G $ is the value $f(V)=sum_{vin V}f(v)$. The minimum weight of an IRDF among all IRDFs on $G$ is called the independent Roman domination number of~$G$. In this paper, we give algorithms for computing the independent Roman domination number of $G$ in $O(|V|)$ time when $G=(V,E)$ is a tree, unicyclic graph or proper interval graph.

Algorithmic aspects of outer independent Roman domination in graphs

Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman dominating function if every vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. If further the set [Formula: see text] is an independent set, then [Formula: see text] is called an outer independent Roman dominating function (OIRDF). Let [Formula: see text] and [Formula: see text]. Then [Formula: see text] is called the outer independent Roman domination number of [Formula: see text]. In this paper, we prove that the decision problem for [Formula: see text] is NP-complete for chordal graphs. We also show that [Formula: see text] is linear time solvable for threshold graphs and bounded tree width graphs. Moreover, we show that the domination and outer independent Roman domination problems are not equivalent in computational complexity aspects.

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

Let $$G=(V,E)$$ be a graph. A function $$f : V \rightarrow \{0, 1, 2\}$$ is an outer independent Roman dominating function (OIRDF) on a graph G if for every vertex $$v \in V$$ with $$f (v) = 0$$ there is a vertex u adjacent to v with $$f (u) = 2$$ and $$\{x\in V:f(x)=0\}$$ is an independent set. The weight of f is the value $$ f(V)=\sum _{v\in V}f(v)$$ . An outer independent total Roman dominating function (OITRDF) f on G is an OIRDF on G such that for every $$v\in V$$ with $$f(v)>0$$ there is a vertex u adjacent to v with $$f (u)>0$$ . The minimum weight of an OIRDF on G is called the outer independent Roman domination number of G, denoted by $$\gamma _{oiR}(G)$$ . Similarly, the outer independent total Roman domination number of G is defined, denoted by $$\gamma _{oitR}(G)$$ . In this paper, we first show that computing $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) is a NP-hard problem, even when G is a chordal graph. Then, for a given proper interval graph $$G=(V,E)$$ we propose an algorithm to compute $$\gamma _{oiR}(G)$$ (respectively, $$\gamma _{oitR}(G)$$ ) in $${\mathcal {O}}(|V| )$$ time.

Complexity Aspects of Variants of Independent Roman Domination in Graphs

For a simple, undirected graph $$G = (V,$$ E), an independent Roman dominating function (IRDF) $$f : V \rightarrow \lbrace 0, 1, 2 \rbrace $$ has the property that every vertex $$v \in V$$ with $$f(v) = 0$$ is adjacent to at least one vertex u, with $$f(u) = 2$$ , and no two vertices assigned positive values are adjacent. An independent Roman $$\lbrace 2 \rbrace $$ -dominating function (IR2DF) is an IRDF f with an additional property that, for every vertex $$v \in V$$ with $$f(v) = 0$$ , there exists at least two vertices $$x, y \in N_G(v)$$ with $$f(x) = f(y) = 1$$ . An independent double Roman dominating function (IDRDF) on G is a function $$f : V \rightarrow \lbrace 0, 1, 2, 3 \rbrace $$ , such that for every vertex $$v \in V$$ , if $$f(v) = 0$$ , then v has at least two neighbors x, y with $$f(x) = f(y) = 2$$ or one neighbor w with $$f(w) = 3$$ , and if $$f(v) = 1$$ , then v must have at least one neighbor w with $$f(w) \ge 2$$ , and no two vertices assigned positive values are adjacent. The weight of an IRDF (IR2DF, IDRDF) f is the sum $$f(V) = \sum _{v \in V}f(v)$$ . Given a graph G and a positive integer k, the independent Roman domination problem (IRDP), independent Roman $$\lbrace 2 \rbrace $$ -domination problem (IR2DP) and independent double Roman domination problem (IDRDP), respectively, is to check whether G has an IRDF, IR2DF, and IDRDF of weight at most k. The IRDP, IR2DP and IDRDP are known to be NP-complete for bipartite graphs. We investigate the complexity of these problems for dually chordal graphs and some subclasses of bipartite graphs, namely, star convex bipartite graphs and comb convex bipartite graphs. We also investigate the complexity of IR2DP and IDRDP for chordal graphs. The minimum independent Roman domination problem (MIRDP), minimum independent Roman $$\lbrace 2 \rbrace $$ -domination problem (MIR2DP), and minimum independent double Roman domination problem (MIDRDP), respectively, are to find an IRDF, IR2DF, and IDRDF of minimum weight in the input graph. We show that MIRDP, MIR2DP, and MIDRDP are linear time solvable for bounded tree-width graphs, chain graphs, and threshold graphs, a subclass of split graphs. We also show that the MIRDP, MIR2DP, and MIDRDP are APX-hard for graphs with bounded degree 4. Finally, we show that the domination problem and IRDP (IR2DP, IDRDP) are not equivalent in computational complexity aspects.

Independent double Roman domination in graphs

For a graph G = (V,E), a double Roman dominating function has the property that for every vertex with f(v) = 0, either there exists a vertex , with f(u) = 3, or at least two neighbors having f(x) = f(y) = 2, and every vertex with value 1 under f has at least a neighbor with value 2 or 3. The weight of a DRDF is the sum . A DRDF f is called independent if the set of vertices with positive weight under f, is an independent set. The independent double Roman domination number is the minimum weight of an independent double Roman dominating function on G. In this paper, we show that for every graph G of order n, and , where and i(G) are the independent 3-rainbow domination, independent Roman domination and independent domination numbers, respectively. Moreover, we prove that for any tree G, .

On the Independent Double Roman Domination in Graphs

An independent double Roman dominating function (IDRDF) on a graph $$G=(V,E)$$ is a function $$f{:}V(G)\rightarrow \{0,1,2,3\}$$ having the property that if $$f(v)=0$$ , then the vertex v has at least two neighbors assigned 2 under f or one neighbor w assigned 3 under f, and if $$f(v)=1$$ , then there exists $$w\in N(v)$$ with $$f(w)\ge 2$$ , such that the set of vertices with positive weight is independent. The weight of an IDRDF is the value $$\sum _{u\in V}f(u)$$ . The independent double Roman domination number $$i_\mathrm{dR}(G)$$ of a graph G is the minimum weight of an IDRDF on G. We continue the study of the independent double Roman domination and show its relationships to both independent domination number (IDN) and independent Roman $$\{2\}$$ -domination number (IR2DN). We present several sharp bounds on the IDRDN of a graph G in terms of the order of G, maximum degree and the minimum size of edge cover. Finally, we show that, any ordered pair (a, b) is realizable as the IDN and IDRDN of some non-trivial tree if and only if $$2a + 1 \le b \le 3a$$ .

Trees with equal Roman {2}-domination number and independent Roman {2}-domination number

A Roman {2}-dominating function (R{2}DF) on a graph G =(V, E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to either at least one vertex v with f(v) = 2 or two vertices v1, v2 with f(v1) = f(v2) = 1. The weight of an R{2}DF f is the value w(f) = ∑u∈Vf(u). The minimum weight of an R{2}DF on a graph G is called the Roman {2}-domination number γ{R2}(G) of G. An R{2}DF f is called an independent Roman {2}-dominating function (IR{2}DF) if the set of vertices with positive weight under f is independent. The minimum weight of an IR{2}DF on a graph G is called the independent Roman {2}-domination number i{R2}(G) of G. In this paper, we answer two questions posed by Rahmouni and Chellali.

Independent Roman domination and 2-independence in trees

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. A Roman dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that every vertex [Formula: see text] for which [Formula: see text] is adjacent to at least one vertex [Formula: see text] for which [Formula: see text]. A Roman dominating function [Formula: see text] is called an independent Roman dominating function if the set of all vertices with positive weights is an independent set. The weight of an independent Roman dominating function [Formula: see text] is the value [Formula: see text]. The independent Roman domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an independent Roman dominating function on [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a 2-independent set of [Formula: see text] if every vertex of [Formula: see text] has at most one neighbor in [Formula: see text]. The maximum cardinality of a 2-independent set of [Formula: see text] is the 2-independence number [Formula: see text]. These two parameters are incomparable in general, however, we show that for any tree [Formula: see text], [Formula: see text] and we characterize all trees attaining the equality.

Independent Roman Domination Number of Graphs

The International Scientific Research Organization for Science, Engineering and Technology (ISROSET) is a Non-Profit Organization; The ISROSET is dedicated to improvement in academic sectors of Science (Chemistry, Bio-chemistry, Zoology, Botany, Biotechnology, Pharmaceutical Science, Bioscience, Bioinformatics, Biometrics, Biostatistics, Microbiology, Environmental Management, Medical Science, Forensic Science, Home Science, Library Science, Material Science Military Science, Physical Science, Physical Education Science, Educational Science, Fisheries, seed technology, Agriculture, Forestry Science, Mathematics, Physics, Statistics and Geology/Earth Science), Computer Science, Engineering and Information Technology, Commerce, Management, Economics Sociology and Social Science.

Independent Roman Domination Stable and Vertex-Critical Graphs

A Roman dominating function (RDF) on a graph $G$ is a function $f: V(G) \rightarrow \{0, 1, 2\}$ for which every vertex assigned 0 is adjacent to a vertex assigned 2. The weight of an RDF is the value $\omega (f) = \sum _{u \in V(G)}f(u)$ . The minimum weight of an RDF on a graph $G$ is called the Roman domination number of $G$ . An RDF $f$ is called an independent Roman dominating function (IRDF) if the set $\{v\in V\mid f(v)\ge 1\}$ is an independent set. The minimum weight of an IRDF on a graph $G$ is called the independent Roman domination number of $G$ and is denoted by $i_{R}(G)$ . A graph $G$ is independent Roman domination stable if the independent Roman domination number of $G$ does not change under removal of any vertex. A graph $G$ is said to be independent Roman domination vertex critical or $i_{R}$ - vertex critical , if for any vertex $v$ in $G$ , $i_{R}(G-v) . In this paper, we characterize independent Roman domination stable trees and we establish upper bounds on the order of independent Roman stable graphs. Also, we investigate the properties of $i_{R}$ - vertex critical graphs. In particular, we present some families of $i_{R}$ -vertex critical graphs and we characterize $i_{R}$ -vertex critical block graphs.

Independent Roman [formula omitted]-domination in graphs

For a graph G=(V,E), a Roman {2}-dominating function (R2DF) f:V→{0,1,2} has the property that for every vertex v∈V with f(v)=0, either there exists an adjacent vertex, a neighbor u∈N(v), with f(u)=2, or at least two neighbors x,y∈N(v) having f(x)=f(y)=1. The weight of a R2DF is the sum f(V)=∑v∈Vf(v). A R2DF f=(V0,V1,V2) is called independent if V1∪V2 is an independent set. The independent Roman {2}-domination number i{R2}(G) is the minimum weight of an IR2DF on G. In this paper, we show that the decision problem associated with i{R2}(G) is NP-complete even when restricted to bipartite graphs. Then we show that for every graph G of order n, 0≤ir2(G)−i{R2}(G)≤n∕5 and 0≤iR(G)−i{R2}(G)≤n∕4, where ir2(G) and iR(G) are the independent 2-rainbow domination and independent Roman domination numbers, respectively. Moreover, we prove that the equality i{R2}(G)=ir2(G) holds for trees.

Outer independent Roman dominating functions in graphs

ABSTRACTA Roman dominating function (RDF) on a graph G is a function satisfying the condition that every vertex u for which is adjacent to at least one vertex v for which . A function is an outer-independent Roman dominating function (OIRDF) on G if f is an RDF and is an independent set. The outer-independent Roman domination number is the minimum weight of an OIRDF on G. In this paper, we initiate the study of the outer-independent Roman domination number in graphs. We first show that determining the number is NP-complete for bipartite graphs. Then we present lower and upper bounds on . Moreover, we characterize graphs with small or large outer-independent Roman domination number.

On the Unique Response Roman Domination Numbers of Graphs

Let G be a graph with vertex set V(G). A function f: V(G) → {0, 1, 2} with the ordered partition (V0, V1, V2) of V(G), where Vi = {V∈V(G) | f(V) = i} for i = 0, 1, 2, is a Roman dominating function if x ∈ V0 implies |N(x)∩V2|≥ 1. It is a unique response Roman function if x ∈ V0 implies |N(x) ≥ V2|≤ 1 and x ∈ V1 ∪ V2 implies that |N(x) ∩ V2| = 0. A function f: V(G) → {0, 1, 2} is a unique response Roman dominating function if it is both a unique response Roman function and a Roman dominating function. The unique response Roman domination number, denoted by uR(G), of G is the minimum weight of a unique response Roman dominating function. In this paper we study the unique response Roman domination of graphs, and provide some graphs whose unique response Roman domination number equals to the independent Roman domination number.