Total Roman domination in graphs

A Roman dominating function (RDF) 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 an RDF is the value f(V (G)) = P u2V (G) f(u). An RDF f in a graph G is independent if no two vertices assigned positive values are adjacent. The Roman domination number R(G) (respectively, the independent Roman domination number iR(G)) is the minimum weight of an RDF (respectively, independent RDF) on G. We say that R(G) strongly equals iR(G), denoted by R(G) ≡ iR(G), if every RDF on G of minimum weight is independent. In this paper we provide a constructive characterization of trees T with R(T) ≡ iR(T).

Suppose = , is a graph. A Roman dominating function (RDF) of is a function: → {0,1,2} such that every vertex for which = 0 has a neighbor with = 2. The weight of an RDF is = Σ ∈ . The Roman domination number (RDN) of a graph , denoted by ℘, is the minimum weight of all possible RDFs. In this study, we define RDF on fuzzy graph (FG). Our purpose is to develop a notion of the RDF and also to present some basic definitions, notations, remarks, and proofs related to RDF on FGs.

For a given graph G without isolated vertex we consider a function f: V(G) rightarrow {0,1,2}. For every iin {0,1,2}, let V_i={vin V(G):; f(v)=i}. The function f is known to be an outer-independent total Roman dominating function for the graph G if it is satisfied that; (i) every vertex in V_0 is adjacent to at least one vertex in V_2; (ii) V_0 is an independent set; and (iii) the subgraph induced by V_1cup V_2 has no isolated vertex. The minimum possible weight omega (f)=sum _{vin V(G)}f(v) among all outer-independent total Roman dominating functions for G is called the outer-independent total Roman domination number of G. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpiński graphs, circulant graphs, and the Cartesian and direct products of complete graphs.

A Roman dominating function (RDF) for a graph is a function satisfying the condition that every vertex u of G for which is adjacent to at least one vertex v of G for which . The weight of a RDF f is the sum , and the minimum weight of a RDF for G is the Roman domination number, of G. A maximal RDF for a graph G is a RDF f such that is not a dominating set of G. The maximal Roman domination number, of a graph G equals the minimum weight of a maximal RDF for G. We first show that determining the number for an arbitrary graph G is NP-complete even when restricted to bipartite or planar graphs. Then, we characterize connected graphs G such that . We also provide a characterization of all trees T of order n such that.

Let G be a graph with no isolated vertex and f:V(G)→{0,1,2} a function. If f satisfies that every vertex in the set {v∈V(G):f(v)=0} is adjacent to at least one vertex in the set {v∈V(G):f(v)=2}, and if the subgraph induced by the set {v∈V(G):f(v)≥1} has no isolated vertex, then we say that f is a total Roman dominating function on G. The minimum weight ω(f)=∑v∈V(G)f(v) among all total Roman dominating functions f on G is the total Roman domination number of G. In this article we study this parameter for the rooted product graphs. Specifically, we obtain closed formulas and tight bounds for the total Roman domination number of rooted product graphs in terms of domination invariants of the factor graphs involved in this product.

On the Roman domination number of a graph

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.

A Roman dominating function (RDF) on a graph G = (V,E) is defined to be a function 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. A set S V is a Restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in . We define a Restrained Roman dominating function on a graph G = (V,E) to be a function 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 and at least one vertex w for which f(w) = 0. The weight of a Restrained Roman dominating function is the value . The minimum weight of a Restrained Roman dominating function on a graph G is called the Restrained Roman domination number of G and denoted by . In this paper, we initiate a study of this parameter.

Some improved inequalities related to Vizing's conjecture

A {\em Roman dominating function} on a graph $G$ is a function‎ ‎$f:V(G)\rightarrow \{0,1,2\}$ satisfying the condition that every‎ ‎vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex‎ ‎$v$ for which $f(v) =2$‎. ‎A {\em restrained Roman dominating}‎ ‎function $f$ is a Roman dominating function if the vertices with label 0 induce‎ ‎a subgraph with no isolated vertex‎. ‎The weight of a restrained Roman dominating function is‎ ‎the value $\omega(f)=\sum_{u\in V(G)} f(u)$‎. ‎The minimum weight of a‎ ‎restrained Roman dominating function of $G$ is called the { \em‎ ‎restrained Roman domination number} of $G$ and denoted by $\gamma_{rR}(G)$.‎ ‎In this paper we establish some sharp bounds for this parameter.‎

The middle graph [Formula: see text] of a graph [Formula: see text] is the graph obtained by subdividing each edge of [Formula: see text] exactly once and joining all these newly introduced vertices of adjacent edges of [Formula: see text]. It is known that the decision problems for Italian domination number [Formula: see text], [Formula: see text]-rainbow domination number [Formula: see text] and Roman domination number [Formula: see text] are NP-complete. Recently, it was proved that [Formula: see text] for a graph [Formula: see text] of order [Formula: see text]. In this paper, we continue to study several domination and domatic numbers in middle graphs. We also obtain closed formulas for domination, secure domination and total domination numbers in middle graphs of rooted product graphs.

Triple Roman domination in graphs

A Roman dominating function (RDF) on a graph G = (V, E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a RDF is the value f(V(G)) = Σu∈V(G) f(u). The minimum weight of a RDF on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number bR(G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E′ ⊆ E(G) for which γR(G - E′) > γR(G). In this paper, we first show that the decision problem for determining bR(G) is NP-hard even for bipartite graphs and then we establish some sharp bounds for bR(G) and characterizes all graphs attaining some of these bounds.

Roman domination was introduced in 2004 by Cockayne, Dreyer, Hedetniemi, and Hedetniemi. If V ( G ) is the vertex set of a graph G, then a function f : V ( G ) → { 0 , 1 , 2 } is a Roman dominating function if every vertex u ∈ V ( G ) for which f ( u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 2 . The Roman domination number of a graph G, denoted by γ R ( G ) , is the minimum weight of a Roman dominating function on G. In this work, we continue to survey what has been done on Roman domination. Mainly, we will present the signed version of some of these functions, and results on the Roman domination number when the structure of the graph is modified by addition of edges/vertices or removing edges/vertices, which are not contained in our previous four survey articles.

In this paper, both the roman domination number and the number of minimum roman dominating sets are found for any rectangular rook’s graph. In a similar fashion, the roman domination number and the number of minimum roman dominating sets are found on the square bishop’s graph for odd board sizes. Also found are the number of minimum total dominating sets associated with the light-colored squares when n ≡ 1(mod12) (with n>1), and same for the dark-colored squares when n ≡ 7(mod12) .