Signed Roman edge k-domination in graphs

Būlitan-i anjuman-i riyāz̤ī-i Īrān., 2024

In 2016, Beeler et al. defined the double Roman domination as a variation of Roman domination. Sometime later, in 2021, Ahangar et al. introduced the concept of [k]-Roman domination in graphs and settled some results on the triple Roman domination case. In 2022, Amjadi et al. studied the quadruple version of this Roman-dominationtype problem. Given any labeling of the vertices of a graph, AN (v) stands for the set of neighbors of a vertex v having a positive label. In this paper we continue the study of the [k]-Roman domination functions ([k]-RDF) in graphs which coincides with the previous versions when 2 ≤ k ≤ 4. Namely, f is a [k]-RDF if f (N [v]) ≥ k +|AN (v)| for all v. We prove that the associate decision problem is NP-complete even when restricted to star convex and comb convex bipartite graphs and we also give sharp bounds and exact values for several classes of graphs.

Let $G=(V,E)$ be a simple graph. For an integer $k\geq 1$, a function $f:V\rightarrow \{0,1,2\}$ is a Roman $k$-tuple dominating function if for any vertex $v$ with $f(v)=0$, there exist at least $k$ vertices $w$ in its neighborhood with $f(w)=2$, and for any vertex $v$ with $f(v)\neq 0$, there exist at least $k-1$ vertices $w$ in its neighborhood with $f(w)=2$. The weight of a Roman $k$-tuple dominating function $f$ of $G$ is the value $f(V)=\sum_{v\in V}f(v)$. The minimum weight of a Roman $k$-tuple dominating function of $G$ is its Roman $k$-tuple domination number. In this paper, we initiate the studying of the Roman $k$-tuple domination number of a graph. Some of our results extend these one given by Cockayne and et al. [Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] for the Roman domination number.

2020

Let G=(V,E) be a finite and simple graph of order n and maximum‎ ‎degree Δ(G)‎. ‎A strong Roman dominating function on a‎ ‎graph G is a function f‎:V (G)→{0‎, ‎1,… ,‎lceil‎ ‎ Δ(G)/2 rceil‎+ ‎1} satisfying the condition that every‎ ‎vertex v for which f(v)=0 is adjacent to at least one vertex u ‎for which‎ f(u) ≤ 1‎+ ‎lceil frac{1}{2}| N(u) ∩ V0| rceil‎, ‎where V0={v ∊ V | f(v)=0}. The minimum of the‎ values sumv∊ V f(v), ‎taken over all strong Roman dominating‎ ‎functions f of G‎, ‎is called the strong Roman domination‎ ‎number of G and is denoted by γStR(G)‎. ‎In this paper we‎ ‎continue the study of strong Roman domination number in graphs‎. ‎In‎ particular‎, ‎we present some sharp bounds for γStR(G) and‎ we determine the strong Roman domination number of some graphs‎.

Iranian Journal of Mathematical Sciences and Informatics, 2020

For any integer k ≥ 1 and any graph G = (V,E) with minimum degree at least k−1, we define a function f : V → {0, 1, 2} as a Roman k-tuple dominating function on G if for any vertex v with f(v) = 0 there exist at least k and for any vertex v with f(v) 6= 0 at least k − 1 vertices in its neighborhood with f(w) = 2. The minimum weight of a Roman k-tuple dominating function f on G is called the Roman k-tuple domination number of the graph where the weight of f is f(V ) = ∑ v∈V f(v). In this paper, we initiate to study the Roman k-tuple domination number of a graph, by giving some sharp bounds for the Roman k-tuple domination number of a garph, the Mycieleskian of a graph, and the corona graphs. Also finding the Roman k-tuple domination number of some known graphs is our other goal. Some of our results extend these one given by Cockayne and et al. [1] in 2004 for the Roman domination number.

Graphs and Combinatorics, 2016

An edge Roman dominating function of a graph G is a function f : E(G) → {0, 1, 2} satisfying the condition that every edge e with f (e) = 0 is adjacent to some edge e ′ with f (e ′) = 2. The edge Roman domination number of G, denoted by γ ′ R (G), is the minimum weight w(f) = e∈E(G) f (e) of an edge Roman dominating function f of G. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if G is a graph of maximum degree ∆ on n vertices, then γ ′ R (G) ≤ ⌈ ∆ ∆+1 n⌉. While the counterexamples having the edge Roman domination numbers 2∆−2 2∆−1 n, we prove that 2∆−2 2∆−1 n + 2 2∆−1 is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of k-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on n vertices is at most 6 7 n, which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain K 2,3 as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs.

Tamkang Journal of Mathematics

Let $G=(V,E)$ be a finite and simple graph of order $n$ and maximumdegree $\Delta$. A signed strong total Roman dominating function ona graph $G$ is a function $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ of $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ is the open neighborhood of $v$ and (ii) every vertex $v$ forwhich $f(v)=-1$ is adjacent to at least one vertex$w$ for which $f(w)\geq 1+\lceil\frac{1}{2}\vert N(w)\cap V_{-1}\vert\rceil$, where$V_{-1}=\{v\in V: f(v)=-1\}$.The minimum of thevalues $\omega(f)=\sum_{v\in V}f(v)$, taken over all signed strongtotal Roman dominating functions $f$ of $G$, is called the signed strong totalRoman domination number of $G$ and is denoted by $\gamma_{ssTR}(G)$.In this paper, we initiate signed strong total Roman domination number of a graph and giveseveral bounds for this parameter. Then, among other results, we determine the signed strong total Roman dominat...

Discussiones Mathematicae Graph Theory, 2013

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value ω(f) = v∈V f (v). The k-distance Roman domination number of a graph G, denoted by γ k R (D), equals the minimum weight of a k-distance Roman dominating function on G. Note that the 1-distance Roman domination number γ 1 R (G) is the usual Roman domination number γ R (G). In this paper, we investigate properties of the k-distance Roman domination number. In particular, we prove that for any connected graph G of order n ≥ k + 2, γ k R (G) ≤ 4n/(2k + 3) and we characterize all graphs that achieve this bound. Some of our results extend these ones given by Cockayne et al. in 2004 and Chambers et al. in 2009 for the Roman domination number.

A Roman dominating function of a graph G =( V, E)i sa function f : V →{ 0, 1, 2} such that every vertex x with f (x)=0 is adjacent to at least one vertex y with f (y) = 2. The weight of a Roman dominating function is defined to be f (V )= P x∈V f (x), and the minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we answer an open problem mentioned in by showing that the Roman domination number of an interval graph can be computed in linear time. We also show that the Roman domination number of a cograph can be computed in linear time. Besides, we show that there are polynomial time algorithms for computing the Roman domination numbers of AT-free graphs and graphs with a d-octopus.

—A Strong Roman dominating function (SRDF) is a function f :V 0,1,2,3 satisfying the condition that every vertex u for which f u   0 is adjacent to at least one vertex v for which f v   3 and every vertex u for which f u  1 is adjacent to at least one vertex v for which f v   2 . The weight of an SRDF is the value        u V f V f u . The minimum weight of an SRDF on a graph G is called the Strong Roman domination numberof G . In this paper, we attempt to verify some properties on SRDF and moreover we present Strong Roman domination number for some special classes of graphs. Also we show that for a tree T with n  3 vertices, l leaves and s support vertices, we have   4 6n l s SR T     and we characterize all trees achieving this bound.

Discrete Mathematics, 2004

A Roman dominating function 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 at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value f (V ) = u∈V f (u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. In this paper we study the graph theoretic properties of this variant of the domination number of a graph.

Discrete Applied Mathematics, 2017

Based on the history that the Emperor Constantine decreed that any undefended place (with no legions) of the Roman Empire must be protected by a "stronger" neighbor place (having two legions), a graph theoretical model called Roman domination in graphs was described. A Roman dominating function for a graph G = (V, E), is a function f : V → {0, 1, 2} such that every vertex v with f (v) = 0 has at least a neighbor w in G for which f (w) = 2. The Roman domination number of a graph is the minimum weight, v∈V f (v), of a Roman dominating function. In this paper we initiate the study of a new parameter related to Roman domination, which we call strong Roman domination number and denote it by γ StR (G). We approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. In particular, we first show that the decision problem regarding the computation of the strong Roman domination number is NP-complete, even when restricted to bipartite graphs. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, we prove that for any tree T of order n ≥ 3, γ StR (T) ≤ 6n/7 and characterize all extremal trees.

European Journal of Pure and Applied Mathematics, 2020

A perfect Roman dominating function on a graph G = (V (G), E(G)) is a function f : V (G) → {0, 1, 2} for which each u ∈ V (G) with f(u) = 0 is adjacent to exactly one vertex v ∈ V (G) with f(v) = 2. The weight of a perfect Roman dominating function f is the value ωG(f) = Pv∈V (G) f(v). The perfect Roman domination number of G is the minimum weight of a perfect Roman dominating function on G. In this paper, we study the perfect Roman domination numbers of graphs under some binary operation

Discussiones Mathematicae Graph Theory

Let G = (V, E) be a graph and let f : V (G) → {0, 1, 2} be a function. A vertex v is said to be protected with respect to f , if f (v) > 0 or f (v) = 0 and v is adjacent to a vertex of positive weight. The function f is a co-Roman dominating function if (i) every vertex in V is protected, and (ii) each v ∈ V with positive weight has a neighbor u ∈ V with f (u) = 0 such that the function f uv : V → {0, 1, 2}, defined by f uv (u) = 1, f uv (v) = f (v) − 1 and f uv (x) = f (x) for x ∈ V \ {v, u}, has no unprotected vertex. The weight of f is ω(f) = v∈V f (v). The co-Roman domination number of a graph G, denoted by γ cr (G), is the minimum weight of a co-Roman dominating function on G. In this paper, we give a characterization of graphs of order n for which co-Roman domination number is 2n 3 or n − 2, which settles Full PDF DMGT Page two open problem in [S. Arumugam, K. Ebadi and M. Manrique, Co-Roman domination in graphs, Proc. Indian Acad. Sci. Math. Sci. 125 (2015) 1-10]. Furthermore, we present some sharp bounds on the co-Roman domination number.

Discussiones Mathematicae Graph Theory

A Roman {2}-dominating function (R2F) is a function f : V → {0, 1, 2} with the property that for every vertex v ∈ 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 γ 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 γ tR2 (G) is NP-complete for bipartite and chordal graphs. Moreover, we show that it is 2 H. Abdollahzadeh Ahangar et al. possible to compute this parameter in linear time for bounded clique-width graphs (including trees).

Opuscula Mathematica, 2013

Let G = (V, E) be a graph and let k be a positive integer.