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.
A \emph{domatic} (\emph{total domatic}) \emph{$k$-coloring} of a graph $G$ is an assignment of $k$ colors to the vertices of $G$ such that each vertex contains vertices of all $k$ colors in its closed neighborhood (neighborhood). The \emph{domatic} (\emph{total domatic}) \emph{number} of $G$, denoted $d(G)$ ($d_t (G)$), is the maximum $k$ for which $G$ has a domatic (total domatic) $k$-coloring. In this paper, we show that for two non-trivial graphs $G$ and $H$, the domatic and total domatic numbers of their Cartesian product $G \cart H$ is bounded above by $\max\{|V(G)|, |V(H)|\}$ and below b…
We say that a vertex-coloring of a graph is a proper k-distance domatic coloring if for each color, every vertex is within distance k from a vertex receiving that color. The maximum number of colors for which such a coloring exists is called the k-distance domatic number of the graph. The problem of determining the k-distance domatic number is motivated by questions about multi-agent networks including arrangements of sensors and robotics. Here, we find the exact k-distance domatic numbers for all grid graphs formed from the Cartesian product of two sufficiently long paths.
For every positive integer $k$, a set $S$ of vertices in a graph $G=(V,E)$ is a $k$-tuple dominating set of $G$ if every vertex of $V-S$ is adjacent to least $k$ vertices and every vertex of $S$ is adjacent to least $k-1$ vertices in $S$. The minimum cardinality of a $k$-tuple dominating set of $G$ is the $k$-tuple domination number of $G$. When $k=1$, a $k$-tuple domination number is the well-studied domination number. We define the $k$-tuple domatic number of $G$ as the largest number of sets in a partition of $V$ into $k$-tuple dominating sets. Recall that when $k=1$, a $k$-tuple domatic nu…
A set $D$ of vertices of a simple graph $G=(V,E)$ is a strong dominating set, if for every vertex $x\in \overline{D}=V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number $γ_{st}(G)$ is defined as the minimum cardinality of a strong dominating set. The strong domatic number of $G$ is the maximum number of strong dominating sets into which the vertex set of $G$ can be partitioned. We initiate the study of the strong domatic number, and we present different sharp bounds on $d_{st}(G)$. In addition, we determine this parameter for some cl…
E. J. Cockayne and S. T. Hedetniemi introduced the concept of domatic number of a graph. B. Zelinka extended the concept to the uniform hypergraphs. Further, B. Zelinka defined the concept of edge-domatic number and total edge-domatic number of a graph. In this paper, we investigate and prove some assertions in connection with vertex domatic number, edge-domatic number and total domatic number of some specific uniform hypergraphs.
Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which partition the vertex set of $G$. In this paper we would like to characterize the cubic graphs with total domatic number at least two.
The domatic number of a graph $G$, denoted $dom(G)$, is the maximum possible cardinality of a family of disjoint sets of vertices of $G$, each set being a dominating set of $G$. It is well known that every graph without isolated vertices has $dom(G) \geq 2$. For every $k$, it is known that there are graphs with minimum degree at least $k$ and with $dom(G)=2$. In this paper we prove that this is not the case if $G$ is $k$-regular or {\em almost} $k$-regular (by ``almost'' we mean that the minimum degree is $k$ and the maximum degree is at most $Ck$ for some fixed real number $C \geq 1$). In thi…
Let $G$ be a graph of order $n$ and size $m$ and let $k\geq 1$ be an integer. A $k$-tuple total dominating set in $G$ is called a $k$-tuple total restrained dominating set of $G$ if each vertex $x\in V(G)-S$ is adjacent to at least $k$ vertices of $V(G)-S$. The minimum number of vertices of a such sets in $G$ are the $k$-tuple total restrained domination number $γ_{\times k,t}^{r}(G)$ of $G$. The maximum number of classes of a partition of $V(G)$ such that its all classes are $k$-tuple total restrained dominating sets in $G$, is called the $k$-tuple total restrained domatic number of $G$. In t…
We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering a conjecture of Goddard and Henning.
A Roman dominating function on a graph $G=(V,E)$ is a function $f: V\to \{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The weight of a Roman dominating function is the value $f(G)=\sum_{u\in V} f(u)$. The Roman domination number of $G$ is the minimum weight of a Roman dominating function on $G$. The Roman bondage number of a nonempty graph $G$ is the minimum number of edges whose removal results in a graph with the Roman domination number larger than that of $G$. This paper determines the exact value of the Roman bo…
The middle graph $M(G)$ of a graph $G$ is the graph obtained by subdividing each edge of $G$ exactly once and joining all these newly introduced vertices of adjacent edges of $G$. A perfect Roman dominating function on a graph $G$ is a function $f : V(G) \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $v$ with $f(v)=0$ is adjacent to exactly one vertex $u$ for which $f(u)=2$. The weight of a perfect Roman dominating function $f$ is the sum of weights of vertices. The perfect Roman domination number is the minimum weight of a perfect Roman dominating function on $G$. In this…
A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. The minimum cardinality of a dominating set for $G$ is called the domination number of $G$. A map $f : V \rightarrow \{0, 1, 2\}$ is a Roman dominating function on a graph $G$ if for every vertex $v$ with $f(v) = 0$, there exists a vertex $u$, adjacent to $v$, such that $f(u) = 2$. The weight of a Roman dominating function is given by $f(V) =\sum_{u\in V}f(u)$. The minimum weight of a Roman dominating function on $G$ is called the Roman domination…
Let $G$ be a graph with vertex set $V(G)$. A function $f:V(G)\rightarrow \{0,1,2\}$ is a Roman dominating function on $G$ if every vertex $v\in V(G)$ for which $f(v)=0$ is adjacent to at least one vertex $u\in V(G)$ such that $f(u)=2$. The Roman domination number of $G$ is the minimum weight $ω(f)=\sum_{x\in V(G)}f(x)$ among all Roman dominating functions $f$ on $G$. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving…
In this paper, we study the double Roman domination number of generalized Sierpiński graphs $S(G,t)$. More precisely, we obtain a bound for the double Roman domination number of $S(G, t)$. We also find the exact value of $γ_{dR}(S(K_{n}, 2))$.
A Roman domination function on a graph G is a function $r:V(G)\to \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $r(u)=0$ is adjacent to at least one vertex $v$ for which $r(v)=2$. The weight of a Roman function is the value $r(V(G))=\sum_{u\in V(G)}r(u)$. The Roman domination number $γ_R(G)$ of $G$ is the minimum weight of a Roman domination function on $G$. "Roman Criticality" has been defined in general as the study of graphs where the Roman domination number decreases when removing an edge or a vertex of the graph. In this paper we give further results in this topic as…
For a graph $G=(V,E)$ of order $n$, a Roman $\{2\}$-dominating function $f:V\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either $v$ is adjacent to a vertex assigned $2$ under $f$, or $v$ is adjacent to least two vertices assigned $1$ under $f$. In this paper, we classify all graphs with Roman $\{2\}$-domination number belonging to the set $\{2,3,4,n-2,n-1,n\}$. Furthermore, we obtain some results about Roman $\{2\}$-domination number of some graph operations.
A map $f : V \rightarrow \{0, 1, 2\}$ is a Roman dominating function on a graph $G=(V,E)$ if for every vertex $v\in V$ with $f(v) = 0$, there exists a vertex $u$, adjacent to $v$, such that $f(u) = 2$. The weight of a Roman dominating function is given by $f(V) =\sum_{u\in V}f(u)$. The minimum weight of a Roman dominating function on $G$ is called the Roman domination number of $G$. In this article we study the Roman domination number of Generalized Sierpiński graphs $S(G,t)$. More precisely, we obtain a general upper bound on the Roman domination number of $S(G,t)$ and we discuss the tightnes…
A $Roman\ domination\ function$ on a graph $G=(V, E)$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The $weight$ of a Roman domination function $f$ 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 $Roman\ domination\ number$ of $G$, denoted by $γ_{R}(G)$. In this paper, we study the {\it Roman domination number} of generalized Petersen graphs P(n,2) and prove that $γ_R(P(n,2)) = \lceil {\frac{8n}{7}}\rceil (n …
For any integer $k\geq 1$ and any graph $G=(V,E)$ with minimum degree at least $k-1$, we define a function $f:V\rightarrow \{0,1,2\}$ 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 $f(v)\neq 0$ 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 $f(V)=\sum_{v\in V}f(v)$. In this paper, we initiate to study the Roman $k$-tuple domination numbe…
A quasi-total Roman dominating function on a graph $G=(V, E)$ is a function $f : V \rightarrow \{0,1,2\}$ satisfying the following: - every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) =2$, and - if $x$ is an isolated vertex in the subgraph induced by the set of vertices labeled with 1 and 2, then $f(x)=1$. The weight of a quasi-total Roman dominating function is the value $ω(f)=f(V)=\sum_{u\in V} f(u)$. The minimum weight of a quasi-total Roman dominating function on a graph $G$ is called the quasi-total Roman domination number of $G$. We introduce th…
A Roman dominating function on a graph $G$ is a labeling $f : V(G) \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The Roman domination number, $γ_R(G)$, of $G$ is the minimum of $Σ_{v\in V (G)} f(v)$ over such functions. The Roman bondage number $b_R(G)$ is the cardinality of a smallest set of edges whose removal from $G$ results in a graph with Roman domination number not equal to $γ_R(G)$. In this paper we obtain upper bounds on $b_{R}(G)$ in terms of (a) the average degree and maximum degree, and (b) Euler characteristic, girth and maximum degr…
For a graph $G=(V,E)$, a double roman dominating function (DRDF) is a function $f : V \longrightarrow \{0, 1, 2,3\}$ having the property that if $f(v)=0$ for some vertex $v$, then $v$ has at least two neighbors assigned $2$ under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$ then $v$ has at least one neighbor $w$ with $f(w) \geq 2$. The weight of a DRDF $f$ is the sum $f (V) =\sum_{u\in V} f (u)$. The minimum weight of a DRDF on a graph $G$ is the double Roman domination number of $G$ and is denoted by $γ_{dR}(G)$. The double roman bondage number of $G$, denoted by $b_{dR}(G)$, is the…
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…
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 $γ_{R}(G)$. 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 $γ_{R}(G-E')>γ_R(G)$. In this paper, we first sho…
For a graph G=(V,E), a restrained double Roman dominating function is a function f:V\rightarrow\{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)\geq2, and at the same time, the subgraph G[V_0] 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)=\sum_{v\in V}f(v), and the minimum weight of a restrained double Roman dominating functi…
A Roman dominating function (RD-function) on a graph $G = (V(G), E(G))$ is a labeling $f : V(G) \rightarrow \{0, 1, 2\}$ such that every vertex with label $0$ has a neighbor with label $2$. The weight $f(V(G))$ of a RD-function $f$ on $G$ is the value $Σ_{v\in V(G)} f (v)$. The {\em Roman domination number} $γ_{R}(G)$ of $G$ is the minimum weight of a RD-function on $G$. The six classes of graphs resulting from the changing or unchanging of the Roman domination number of a graph when a vertex is deleted, or an edge is deleted or added are considered. We consider relationships among the classes…
For a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2\}$ is called Roman dominating function (RDF) if for any vertex $v$ with $f(v)=0$, there is at least one vertex $w$ in its neighborhood with $f(w)=2$. The weight of an RDF $f$ of $G$ is the value $f(V)=\sum_{v\in V}f(v)$. The minimum weight of an RDF of $G$ is its Roman domination number and denoted by $γ_ R(G)$. In this paper, we first show that $γ_{R}(G)+1\leq γ_{R}(μ(G))\leq γ_{R}(G)+2$, where $μ(G)$ is the Mycielekian graph of $G$, and then characterize the graphs achieving equality in these bounds. Then for any positive integer $m$,…
For a graph $G = (V, E)$, a Roman dominating function $f : V \rightarrow \{0, 1, 2\}$ has the property that every vertex $v \in V $with $f (v) = 0$ has a neighbor $u$ with $f (u) = 2$. The weight of a Roman dominating function $f$ is the sum $f (V) = \cup_{v\in V} f (v)$, and the minimum weight of a Roman dominating function on $G$ is the Roman domination number $γ_R(G)$ of $G$. The Roman bondage number $b_R(G)$ of $G$ is the minimum cardinality of all sets $F \subseteq E$ for which $γ_R(G - F) > γ_R(G)$. A graph $G$ is in the class $\mathcal{R}_{UVR}$ if the Roman domination number remains un…
The Roman dominating function on a graph $G=(V,E)$ is a function $f: V\rightarrow\{0,1,2\}$ such that each vertex $x$ with $f(x)=0$ is adjacent to at least one vertex $y$ with $f(y)=2$. The value $f(G)=\sum\limits_{u\in V(G)} f(u)$ is called the weight of $f$. The Roman domination number $γ_{\rm R}(G)$ is defined as the minimum weight of all Roman dominating functions. This paper defines the Roman bondage number $b_{\rm R}(G)$ of a nonempty graph $G=(V,E)$ to be the cardinality among all sets of edges $B\subseteq E$ for which $γ_{\rm R}(G-B)>γ_{\rm R}(G)$. Some bounds are obtained for $b_{\rm …
In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination number, and the total restrained domination number is equal to the total domination number. A number of open problems are posed.
For a given graph $G$ without isolated vertex we consider a function $f: V(G) \rightarrow \{0,1,2\}$. For every $i\in \{0,1,2\}$, let $V_i=\{v\in V(G):\; f(v)=i\}$. 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 $V_1\cup V_2$ has no isolated vertex. The minimum possible weight $ω(f)=\sum_{v\in V(G)}f(v)$ among all outer-independent total Roman dominating functions for $G$ is ca…
Given a graph $G=(V,E)$, a function $f:V\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then there exist $ v_{1},v_{2}\in N(v)$ such that $f(v_{1})=f(v_{2})=2$ or there exists $ w \in N(v)$ such that $f(w)=3$, and if $f(v)=1$, then there exists $ w \in N(v)$ such that $f(w)\geq 2$ is called a double Roman dominating function (DRDF). The weight of a DRDF $f$ is the sum $f(V)=\sum_{v\in V}f(v)$. The double Roman domination number, $γ_{dR}(G)$, is the minimum among the weights of DRDFs on $G$. In this paper, we study the impact of some graph operations, such as cartesian product, a…
Given a graph $G$ without isolated vertices, a total Roman dominating function for $G$ is a function $f : V(G)\rightarrow \{0,1,2\}$ such that every vertex with label 0 is adjacent to a vertex with label 2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number $γ_{tR}(G)$ of $G$ is the smallest possible value of $\sum_{v\in V(G)}f(v)$ among all total Roman dominating functions $f$. The total Roman domination number of the direct product $G\times H$ of the graphs $G$ and $H$ is studied in this work. Specifically, several r…
An edge Roman dominating function of a graph $G$ is a function $f\colon E(G) \rightarrow \{0,1,2\}$ 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 $w(f) = \sum_{e\in E(G)} f(e)$ 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 $Δ$ on $n$ vertices, then $γ_R'(G) \le \lceil \fracΔ{Δ+1} n \rceil$. While t…
For a graph $G$, let $γ_{r2}(G)$ and $γ_R(G)$ denote the $2$-rainbow domination number and the Roman domination number, respectively. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) proved $γ_{r2}(G)+γ_R(G)\leq \frac{6}{4}n(G)$ for a connected graph $G$ of order $n(G)$ at least $3$. Furthermore, they conjectured $γ_{r2}(G)+γ_R(G)\leq \frac{4}{3}n(G)$ for a connected graph $G$ of minimum degree at least $2$ that is distinct from $C_5$. We characterize all extremal graphs for their inequality and prove th…
A \emph{perfect Roman dominating function} (PRDF) on a graph $G = (V, E)$ is a function $f : V \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$. The weight of a PRDF is the value $w(f) = \sum_{u \in V}f(u)$. The minimum weight of a PRDF on a graph $G$ is called the \emph{perfect Roman domination number $γ_R^p(G)$} of $G$. A graph $G$ is perfect Roman domination domination stable if the perfect Roman domination number of $G$ remains unchanged under the removal of any vertex. In this paper, we …
Let $G=(V(G),E(G))$ be a simple graph. A restrained double Roman dominating function (RDRD-function) of $G$ is a function $f: V(G) \rightarrow \{0,1,2,3\}$ satisfying the following properties: if $f(v)=0$, then the vertex $v$ has at least two neighbours assigned 2 under $f$ or one neighbour $u$ with $f(u)=3$; and if $f(v)=1$, then the vertex $v$ must have one neighbor $u$ with $f(u) \geq 2$; the induced graph by vertices assigned 0 under $f$ contains no isolated vertex. The weight of a RDRD-function $f$ is the sum $f(V)=\sum_{v \in V(G)} f(v)$, and the minimum weight of a RDRD-function on $G$ …
An independent double Roman dominating function (IDRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ has at least two neighbors assigned $2$ under $f$ or one neighbor $w$ with assigned $3$ under $f$, and if $f(v)=1$, then there exists $w\in N(v)$ with $f(w)\geq2$ such that the positive weight vertices are independent. The weight of an IDRDF is the value $\sum_{u\in V}f(u)$. The independent double Roman domination number $i_{dR}(G)$ of a graph $G$ is the minimum weight of an IDRDF on G. We initiate the study of the …
Let $G=(V,E)$ be a graph of order $n$ and let $γ_{R}(G)$ 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 $δ\geq 2,$ which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$% -cycles, then $γ_{R}(G)\leq \frac{(4k+8)n}{6k+11}$. 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…
In this paper we deal with the calculation of the signed (total) Roman domination numbers, $γ_{sR}$ and $γ_{stR}$ respectively, on a few classes of planar graphs from the literature. We give proofs for the exact values of the numbers $γ_{sR}(A_n)$ and $γ_{sR}(R_n)$ as well as the numbers $γ_{stR}(S_n)$ and $γ_{stR}(T_n)$. For some other classes of planar graphs, such as $Q_n$, %$S_n"$ and $T_n"$, lower and upper bounds on $γ_{sR}$ are calculated and proved. %We give some open problems on the exact values of $γ_{sR}$ and $γ_{stR}$ for some classes of planar graphs.
