Some progress on the mixed Roman domination in graphs

European Journal of Pure and Applied Mathematics

Let G be a connected graph. A function f : V (G) → {0, 1, 2, 3} is a double Roman dominating function of G if for each v ∈ V (G) with f(v) = 0, v has two adjacent vertices u and w for which f(u) = f(w) = 2 or v has an adjacent vertex u for which f(u) = 3, and for each v ∈ V (G) with f(v) = 1, v is adjacent to a vertex u for which either f(u) = 2 or f(u) = 3. The minimum weight ωG(f) = P v∈V (G) f(v) of a double Roman dominating function f of G is the double Roman domination number of G. In this paper, we continue the study of double Roman domination introduced and studied by R.A. Beeler et al. in [2]. First, we characterize some double Roman domination numbers with small values in terms of the domination numbers and 2-domination numbers. Then we determine the double Roman domination numbers of the join, corona, complementary prism and lexicographic product of graphs.

—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.

RAIRO - Operations Research

Let G be a simple, undirected graph. A function g : V(G) → {0, 1, 2, 3} having the property that ∑v∈NG(u)g(v)≥3, if g(u) = 0, and ∑v∈NG(u)g(v)≥2, if g(u) = 1 for any vertex u ∈ G, where NG(u) is the set of vertices adjacent to u in G, is called a Roman {3}-dominating function (R3DF) of G. The weight of a R3DF g is the sum g(V)=∑v∈Vg(v). The minimum weight of a R3DF is called the Roman {3}-domination number and is denoted by γ{R3}(G). Given a graph G and a positive integer k, the Roman {3}-domination problem (R3DP) is to check whether G has a R3DF of weight at most k. In this paper, first we show that the R3DP is NP-complete for chordal graphs, planar graphs and for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. The minimum Roman {3}-domination problem (MR3DP) is to find a R3DF of minimum weight in the input graph. We show that MR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a s...

RAIRO - Operations Research

For a graph G = (V, E), a restrained double Roman dominating function is a function f : V → {0, 1, 2, 3} having the property that if f(v) = 0, then the vertex v must have at least two neighbors assigned 2 under f or one neighbor w with f(w) = 3, and if f(v) = 1, then the vertex v must have at least one neighbor w with f(w) ≥ 2, and at the same time, the subgraph G[V0] which includes vertices with zero labels has no isolated vertex. The weight of a restrained double Roman dominating function f is the sum f(V) = ∑v∈V f(v), and the minimum weight of a restrained double Roman dominating function on G is the restrained double Roman domination number of G. We initiate the study of restrained double Roman domination with proving that the problem of computing this parameter is NP-hard. Then we present an upper bound on the restrained double Roman domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. We study the restrained doub...

Discrete Applied Mathematics, 2018

A signed double Roman dominating function (SDRDF) on a graph G = (V , E) is a function f : V (G) → {−1, 1, 2, 3} such that (i) every vertex v with f (v) = −1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f (w) = 3, (ii) every vertex v with f (v) = 1 is adjacent to at least one vertex w with f (w) ≥ 2 and (iii) ∑ u∈N[v] f (u) ≥ 1 holds for any vertex v. The weight of an SDRDF f is ∑ u∈V (G) f (u), the minimum weight of an SDRDF is the signed double Roman domination number γ sdR (G) of G. In this paper, we prove that the signed double Roman domination problem is NP-complete for bipartite and chordal graphs. We also prove that for any tree T of order n ≥ 2, −5n+24 9 ≤ γ sdR (T) ≤ n and we characterize all trees attaining each bound.

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.

2005

A Roman dominating function of a graph G = (V,E) is a 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) = ∑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 [2] 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.

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, 2008

A Roman dominating function of a graph G = (V, E) is a 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 ) = 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 first answer an open question mentioned in [E.J. Cockayne, P.A. Dreyer Jr., S.M. Hedetniemi, S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11-22] by showing that the Roman domination number of an interval graph can be computed in linear time. We then show that the Roman domination number of a cograph (and a graph with bounded cliquewidth)

Proceedings - Mathematical Sciences, 2015

Let G = (V , E) be a graph and let f : V →{0, 1, 2} be a function. A vertex u is said to be protected with respect to f if f(u) > 0orf(u) = 0andu is adjacent to a vertex with positive weight. The function f is a co-Roman dominating function (CRDF) if: (i) every vertex in V is protected, and (ii) each v ∈ V with f(v) > 0 has a neighbor u ∈ V with f(u) = 0 such that the function f vu : V →{ 0, 1, 2}, defined by f vu (u) = 1, f vu (v) = f(v) − 1a n df vu (x) = f(x) for x ∈ V \{u, v} has no unprotected vertex. The weight of f is w(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 initiate a study of this parameter, present several basic results, as well as some applications and directions for further research. We also show that the decision problem for the co-Roman domination number is NP-complete, even when restricted to bipartite, chordal and planar graphs.