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A Roman $\{2\}$-dominating function (R2F) is a function $f:V\rightarrow \{0,1,2\}$ with the property that for every vertex $v\in V$ 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 $\gamma_{tR2}(G).$ 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 $\gamma_{tR2}(G)$ is NP-complete for bipartite and chordal graphs. {Moreover, we show that it is possible to compute this parameter in linear time for bounded clique-width graphs (including tres).}

Let $\{0,1,\dots, t\}$ be abbreviated by $[t].$ A double Roman dominating function (DRDF) on a graph $\Gamma=(V,E)$ is a map $l:V\rightarrow [3]$ satisfying \textrm{(i)} if $l(r)=0$ then there must be at least two neighbors labeled 2 under $l$ or a neighbor $r'$ with $l(r')=3$; and \textrm{(ii)} if $l(r)=1$ then $r$ must be adjacent to a vertex $r'$ such that $l(r')\geq2$. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on $\Gamma$ if the set of vertices labeled $0$ induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its map values over all vertices, and the outer independent total Roman dominating number $\gamma_{tdR}^{oi}(\Gamma)$ is the minimum weight of an OITDRDF on $\Gamma$. First, we prove that the problem of determining $\gamma _{tdR}^{oi}(\Gamma)$ is NP-complete for bipartite and chordal graphs, after that, we prove that it is solvable in linear time when we are restricting to bounded clique-width graphs. Moreover, we present some tight bounds on $\gamma _{tdR}^{oi}(\Gamma)$ as well as the exact values for several graph families.

A maximal double Roman dominating function (MDRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ such that \textrm{(i) }every vertex $v$ with $f(v)=0$ is adjacent to least two vertices { assigned $2$ or to at least one vertex assigned $3,$} \textrm{(ii) }every vertex $v$ with $f(v)=1$ is adjacent to at least one { vertex assigned $2$ or $3$} and \textrm{(iii) }the set $\{w\in V|~f(w)=0\}$ is not a dominating set of $G $. The weight of a MDRDF is the sum of its function values over all vertices, and the maximal double Roman domination number $\gamma _{dR}^{m}(G) $ is the minimum weight of an MDRDF on $G$. {In this paper, we initiate the study of maximal double Roman domination. We first show that the problem of determining }$\gamma _{dR}^{m}(G)$ {is NP-complete for bipartite, chordal and planar graphs. But it is solvable in linear time for bounded clique-width graphs including trees, cographs and distance-hereditary graphs. Moreover, we establish various relationships relating }$\gamma _{dR}^{m}(G)$ to some domination parameters. {For the class of trees, we show that for every tree }$T$ {of order }$n\geq 4,$ $\gamma _{dR}^{m}(T)\leq \frac{5}{4}n$ {and we characterize all trees attaining the bound. Finally, the exact values of }$\gamma _{dR}^{m}(G) $ {are given for paths and cycles.

Let $G=(V,E)$ be a graph of order $n$ and let $\gamma _{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of order $n\geq 6k+9$, minimum degree $\delta \geq 2,$ which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$% -cycles, then $\gamma _{R}(G)\leq \frac{(4k+8)n}{6k+11}$. This bound is an improvement of the bounds given in [E.W. Chambers, B. Kinnersley, N. Prince, and D.B. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575--1586] when $k=0,$ {and [S. Bermudo, On the differential and Roman domination number of a graph with minimum degree two, Discrete Appl. Math. 232 (2017), 64--72] when }$k=1.$ Moreover, using the Gallai-type result involving the Roman domination number and the differential of graphs established by Bermudo et al. stating that $\gamma _{R}(G)+\partial (G)=n$, we have $\partial (G)\geq \frac{(2k+3)n}{6k+11},$ thereby settling the conjecture of Bermudo posed in the second paper.

We continue the study of restrained double Roman domination in graphs. For a graph $G=\big{(}V(G),E(G)\big{)}$, a double Roman dominating function $f$ is called a restrained double Roman dominating function (RDRD function) if the subgraph induced by $\{v\in V(G)\mid f(v)=0\}$ has no isolated vertices. The restrained double Roman domination number (RDRD number) $\gamma_{rdR}(G)$ is the minimum weight $\sum_{v\in V(G)}f(v)$ taken over all RDRD functions of $G$. We first prove that the problem of computing $\gamma_{rdR}$ is NP-hard even for planar graphs, but it is solvable in linear time when restricted to bounded clique-width graphs such as trees, cographs and distance-hereditary graphs. Relationships between $\gamma_{rdR}$ and some well-known parameters such as restrained domination number $\gamma_{r}$, domination number $\gamma$ and restrained Roman domination number $\gamma_{rR}$ are investigated in this paper by bounding $\gamma_{rdR}$ from below and above involving $\gamma_{r}$, $\gamma$ and $\gamma_{rR}$ for general graphs, respectively. We prove that $\gamma_{rdR}(T)\geq n+2$ for any tree $T\neq K_{1,n-1}$ of order $n\geq2$ and characterize the family of all trees attaining the lower bound. The characterization of graphs with small RDRD numbers is given in this paper.

The forgotten topological index of a graph $G$, denoted by $F(G)$, is defined as the sum of weights $d(u)^{2}+d(v)^{2}$ over all edges $uv$ of $G$ , where $d(u)$ denotes the degree of a vertex $u$. In this paper, we give sharp upper bounds of the F-index (forgotten topological index) over bicyclic graphs, in terms of the order and maximum degree.

Let $k$ be a positive integer. A {\em Roman $k$-dominating function} on a graph $G$ is a labeling $f:V (G)\longrightarrow \{0, 1, 2\}$ such that every vertex with label 0 has at least $k$ neighbors with label 2. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct Roman $k$-dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 2k$ for each $v\in V(G)$, is called a {\em Roman $(k,k)$-dominating family} (of functions) on $G$. The maximum number of functions in a Roman $(k,k)$-dominating family on $G$ is the {\em Roman $(k,k)$-domatic number} of $G$, denoted by $d_{R}^k(G)$. Note that the Roman $(1,1)$-domatic number $d_{R}^1(G)$ is the usual Roman domatic number $d_{R}(G)$. In this paper we initiate the study of the Roman $(k,k)$-domatic number in graphs and we present sharp bounds for $d_{R}^k(G)$. In addition, we determine the Roman $(k,k)$-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.

Consider a finite and simple graph $G=(V,E)$ with maximum degree $\Delta$. A strong Roman dominating function over the graph $G$ is understood as a map $f : V (G)\rightarrow \{0, 1,\ldots , \left\lceil \frac{\Delta}{2}\right\rceil+ 1\}$ which carries out the condition stating that all the vertices $v$ labeled $f(v)=0$ are adjacent to at least one another vertex $u$ that satisfies $f(u)\geq 1+ \left\lceil \frac{1}{2}\vert N(u)\cap V_0\vert \right\rceil$, such that $V_0=\{v \in V \mid f(v)=0 \}$ and the notation $N(u)$ stands for the open neighborhood of $u$. The total version of one strong Roman dominating function includes the additional property concerning the not existence of vertices of degree zero in the subgraph of $G$, induced by the set of vertices labeled with a positive value. The minimum possible value for the sum $\omega(f)=f(V)=\sum_{v\in V} f(v)$ (also called the weight of $f$), taken amongst all existent total strong Roman dominating functions $f$ of $G$, is called the total strong Roman domination number of $G$, denoted by $\gamma_{StR}^t(G)$. This total and strong version of the Roman domination number (for graphs) is introduced in this research, and the study of its mathematical properties is therefore initiated. For instance, we establish upper bounds for such parameter, and relate it with several parameters related to vertex domination in graphs, from which we remark the standard domination number, the total version of the standard domination number and the (strong) Roman domination number. In addition, among other results, we show that for any tree $T$ of order $n(T)\ge 3$, with maximum degree $\Delta(T)$ and $s(T)$ support vertices, $\gamma_{StR}^t(T)\ge \left\lceil \frac{n(T)+s(T)}{\Delta(T)}\right\rceil+1$., Comment: 19 pages

For a function $f : V(G ) \rightarrow \{0, 1, 2\}$ we denote by $V_i$ the set of vertices to which the value $i$ is assigned by $f$, i.e. $V_i = \{ x \in V (G ) : f(x ) = i \}$. If a function $f: V(G) \rightarrow \{0,1,2\}$ satisfying the condition that $V_i$ is independent for $i \in \{1,2\}$ and every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = i$ for each $i \in \{1,2\}$, then $f$ is called a 2-rainbow independent dominating function (2RiDF). The weight $w(f)$ of a 2RiDF $f$ is the value $w(f) = |V_1|+|V_2|$. The minimum weight of a 2RiDF on a graph $G$ is called the \emph{2-rainbow independent domination number} of $G$. A graph $G$ is 2-rainbow independent domination stable if the 2-rainbow independent domination number of $G$ remains unchanged under removal of any vertex. In this paper, we characterize 2-rainbow independent domination stable trees and we study the effect of edge removal on 2-rainbow independent domination number in trees.

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 $f:V\rightarrow \{0,1,2\}$ 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, $\sum_{v\in V}f(v)$, 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 $\gamma_{StR}(G)$. 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 $n\ge 3$, $\gamma_{StR}(T)\le 6n/7$ and characterize all extremal trees., Comment: 23 pages

A Roman dominating function on a graph $G=(V,E)$ is a function $f:V\rightarrow\{0,1,2\}$ such that every vertex $v\in V$ with $f(v)=0$ has at least one neighbor $u\in V$ with $f(u)=2$. The weight of a Roman dominating function is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a Roman dominating function on a graph $G$ is called the Roman domination number, denoted by $\gamma_{R}(G)$. The Roman bondage number $b_{R}(G)$ of a graph $G$ with maximum degree at least two is the minimum cardinality of all sets $E'\subseteq E(G)$ for which $\gamma_{R}(G-E')>\gamma_R(G)$. In this paper, we first show that the decision problem for determining $b_{\rm R}(G)$ is NP-hard even for bipartite graphs and then we establish some sharp bounds for $b_{\rm R}(G)$ and characterizes all graphs attaining some of these bounds., Comment: 15 pages, 35 references