Signed Roman edge domination numbers in graphs

A Roman dominating function on a graph G is a function f:V(G) → {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 f is the sum, ΣuV(G) f(u), of the weights of the vertices. The Roman domination number is the minimum weight of a Roman dominating function in G. A total Roman domination function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The total Roman domination number is the minimum weight of a total Roman domination function on G. We establish lower and upper bounds on the total Roman domination number. We relate the total Roman domination to domination parameters, including the domination number, the total domination number and Roman domination number.

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.

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.

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.‎

On the Roman domination number of a graph

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) .

Let γPm □ Cn denote the domination number of the cylindrical grid graph formed by the Cartesian product of the graphs Pm, the path of length m, m≥2, and the graph Cn, the cycle of length n, n≥3. In this paper we propose methods to find the domination numbers of graphs of the form Pm □ Cn with n≥3 and m=5 and propose tight bounds on domination numbers of the graphs P6 □ Cn, n≥3. Moreover, we provide rough bounds on domination numbers of the graphs Pm □ Cn, n≥3 and m≥7. We also point out how domination numbers and minimum dominating sets are useful for wireless sensor networks.

A Roman dominating function of a graph G is a function f$: V(G) \to \{0, 1, 2\}$ such that whenever $f(v)=0$, there exists a vertex u adjacent to v such that $f(u) = 2$. The weight of f is $w(f) = \sum_{v \in V(G)} f(v)$. The Roman domination number $\gamma_R(G)$ of G is the minimum weight of a Roman dominating function of G. Chambers, Kinnersley, Prince, and West [SIAM J. Discrete Math., 23 (2009), pp. 1575–1586] conjectured that $\gamma_R(G) \le \lceil 2n/3 \rceil$ for any 2-connected graph G of n vertices. This paper gives counterexamples to the conjecture and proves that $\gamma_R(G) \le \max\{\lceil 2n/3 \rceil, 23n/34\}$ for any 2-connected graph G of n vertices. We also characterize 2-connected graphs G for which $\gamma_R(G) = 23n/34$ when $23n/34 > \lceil 2n/3 \rceil$.

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.

Let $$D=(V,A)$$ be a digraph. A Roman dominating function of a digraph D is a function f : $$V\longrightarrow \{0,1,2\}$$ such that every vertex u for which $$f(u)=0$$ has an in-neighbor v for which $$f(v)=2$$ . The weight of a Roman dominating function is the value $$f(V)=\sum _{u\in V}f(u)$$ . The minimum weight of a Roman dominating function of a digraph D is called the Roman domination number of D, denoted by $$\gamma _{R}(D)$$ . In this paper, we characterize some special classes of oriented graphs, namely out-regular oriented graphs and tournaments satisfying $$\gamma _{R}(D)=n-\Delta ^{+}(D)+1$$ . Moreover, we characterize digraphs D for which the equality $$\gamma _{R}(D)+\gamma _{R}(\overline{D})=n+3$$ holds, where $$\overline{D}$$ is the complement of D. Finally, we prove that the problem of deciding whether an oriented graph D satisfies $$\gamma _{R}(D)=n-\Delta ^{+}(D)+1$$ is CO- $$\mathcal {NP}$$ -complete.

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.

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).

A Roman dominating function on a graph G is a function f : V(G) → {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 $${f (V(G)) = \sum_{u\in V(G)} f (u)}$$. The Roman domination number, γ R (G), of G is the minimum weight of a Roman dominating function on G. The Roman bondage number b R (G) of a graph G with maximum degree at least two is the minimum cardinality of all sets $${E^{\prime} \subseteq E(G)}$$ for which γ R (G − E′) > γ R (G). In this paper we present different bounds on the Roman bondage number of planar graphs.