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\title[Italian Domination Number of Some Snark Family]{\large Italian Domination Number of Some Snark Family}


\author[AMRITHA PRAKASH]{{\bf AMRITHA PRAKASH}\protect\orcidicon{0009-0005-2896-0964}}
\author[RAGUKUMAR P]{{\bf RAGUKUMAR P.}\protect\orcidicon{0000-0002-2923-3268}}



\address{AMRITHA PRAKASH  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Department of Mathematics,
\newline\hphantom{iii} School of Advanced Sciences,
\newline\hphantom{iii} Vellore Institute of Technology,
\newline\hphantom{iii} 632 014, Vellore, India}%
\email{\textcolor{blue}{amrithasoubhagya10@gmail.com }}%

\address{RAGUKUMAR PANDURANGAN % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Department of Mathematics,
\newline\hphantom{iii} School of Advanced Sciences,
\newline\hphantom{iii} Vellore Institute of Technology,
\newline\hphantom{iii} 632 014, Vellore, India}%
\email{\textcolor{blue}{ragukumar2003@gmail.com }}%



\thanks{\sc Amritha Prakash,
Italian Domination Number of Some Snark Family}
%\thanks{\copyright \ 2023 Ivanov I.I.}
\thanks{\rm The work is financially supported by Vellore Institute of Technology , Vellore}
\thanks{\it Received , Published }%


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\maketitle {\small

\vspace{-12pt}
\centerline{{\it Communicated by} {\sc }
%({\it filled by Editorial board})
}

\bigskip
\bigskip

\begin{quote}
\noindent{\bf Abstract:} For a graph	$G = (V, E)$, an Italian dominating function (IDF) $f : V \rightarrow \{0, 1, 2\}$ 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 at least two vertices assigned 1 under $f$. The weight of an Italian dominating function is the $\sum_{v\in V} f(v)$, and the minimum weight of a Italian dominating function $f$ is the Italian domination number. Determining the Italian domination number of a graph is a well discussed  NP-complete problem. Finding an exact value of Italian domination number is NP-complete even for bipartite graphs. In this work we identifies the Italian domination number of some families of snarks such as flower snarks, Loupekine snarks, Goldberg snarks, Watkin snark and Szekeres snark.\medskip

\noindent{\bf Keywords:} Domination, Italian domination number, 
Roman domination, Regular graphs, Snark.
 \end{quote}
}

\bigskip

	\section{Introduction}
	
	All graphs studied here are simple, finite and undirected. For additional notations and terminologies not given here we refer to \cite{Bondy,fundom,west}. Let $G =(V(G),E(G))$ be a graph with vertex set $V(G)$ of order $n = |V(G)|$ and edge set $E(G)$. The open neighborhood $N(v)$ of a vertex $v$ consists of the vertices adjacent to $v$, while the closed neighborhood is $N[v] = N(v)\cup \{v\}$. The degree of $v$ ($deg(v)$) is the cardinality of its open neighborhood. The maximum degree of $G$ is the number $\Delta(G) = \text{max}\{deg(v):v \in V(G)\}$. A graph in which all of its vertices are of degree three is called a cubic graph.  A dominating set is a set $D\subseteq V(G)$ such that any vertex of G is in D or has a neighbor in D. The minimum cardinality of a domination set of G is called the domination number of G, denoted by $\gamma(G)$.\\
	The domination of graphs has been extensively researched in the literature \cite{fundom,dom2,dom3}, resulting in different variations and generalizations. One of these variations, with historical significance, came to be called Roman domination.  Stewart \cite{stewart} introduced the concept of Roman domination on graphs, which was further developed by Cockayne et al. \cite{roman}. This concept was motivated by the military problem faced by Emperor Constantine the Great in the fourth century A.D., when the Roman Empire was under attack. The Italian domination is another significant variant of Roman domination that was first put forward by Chellali et al. \cite{roman2} as Roman $\{2\}$- domination.  Based on the defense plan of Roman empire, every location that lacks a legion must have a surrounding locations with two legion or at least two neighboring locations that each carry one legion. Formally, Italian-dominating function $f : V \rightarrow \{0, 1, 2\}$ has the property that for every vertex $v \in V$, with $f(v) = 0$ either there is a vertex $u \in N(v)$ with $f(u) = 2$, or at least two vertices $u, w \in N(v)$ with $f(u) = f(w) = 1$. The weight of an Italian dominating function is the $\sum_{v\in V} f(v)$ and the minimum weight of an Italian dominating function $f$ is the Italian domination number, denoted by $\gamma_{I}(G)$. Figure \ref{fig1} illustrates an Italian dominating function assigned for the Petersen graph. 
	
	
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		\caption{ IDF assigned on Petersen graph with $w(f)=4.$}\label{fig1}
	\end{figure}
	
	
	Italian domination number was connected with other domination parameters by  Chellali et al. \cite{roman2}  and also showed that Italian domination is NP-complete for bipartite graphs. Henning et al. \cite{tree} characterize the trees $T$ with $\gamma(T) +1 = \gamma_{I}(T)$ and $\gamma_{I}(T) = 2 \gamma (T)$ in 2017. Subsequently, in 2021 various results based on binary operations in Italian domination are presented, particularly for Cartesian products and rooted products \cite{cartesian,rooted}. Graphs with restricted maximum degree are frequently taken into consideration in the study of most graph parameters. The Italian domination number has been calculated for connected graphs with a maximum degree of two, which are either paths or cycles \cite{roman2}. The next stage is to explore graphs with a maximum degree of three. So far, the Italian domination number for a cubic graph has only been found for generalized Petersen graphs \cite{genpetersen}. Being snarks are cubic graphs and other forms of domination have been researched, no one has explored the Italian domination number of this particular graph classes. This motivated us to explore the Italian domination number of snark families.
	In 1880, Tait demonstrated that the Four-Color Theorem is identical to the statement that every bridgeless cubic planar graph has a 3-edge coloring \cite{tait}, beginning off the study of snarks. Snarks are non-Hamiltonian and remain interesting in graph theory because they serve as minimal counter examples to certain well-known conjectures. \\
	This study discusses the Italian domination over certain members of the snark family. In Section 2, we present some preliminary findings. In Section 3, we describe our key findings on the Italian domination number of snark families such as flower snarks, Goldberg snarks, Loupekine snarks, Watkin snark, and Szekeres snark.
	
	\section{Preliminary results}
	In this section, we illustrate some basic bounds and theorems on  Italian domination number of a graph.
	\begin{theorem} \cite{roman2}\label{lb}
		If $G$ is a connected graph of order $n$ and maximum degree $\Delta(G) = \Delta$, then $\gamma_{I}(G) \geq \frac{2n}{(\Delta + 2)}$.
	\end{theorem}
	
	\begin{theorem}\cite{roman2} For every graph $G, \gamma(G) \leq \gamma_{I}(G) \leq 2 \gamma (G)$.
	\end{theorem}
	
	\begin{theorem}\cite{roman2} \label{thmpc}
		For the classes of paths $P_n$ and cycles $C_n$,
		$\gamma_{I}(P_n) = \lceil\frac{n+1}{2}\rceil $ and $\gamma_{I}(C_n) = \lceil \frac{n}{2}\rceil$.
	\end{theorem}
	\begin{proposition}\cite{digraph}\label{pp1}
		If $G$ is a graph of order $n$, then $\gamma_{I}(G) = n$ if and only if $\Delta(G)
		\leq1$.
	\end{proposition}
	\begin{proposition}\cite{digraph}\label{pp2}
		If $G$ is a graph of order $n \geq 2$, then $\gamma_{I}(G) = 2$ if and only if
		$\Delta(G)=n-1 $ or there exist two different vertices  $u$ and $v$ such that $N(u) \cap N(v)
		=V(G)\backslash\{u,v\}$.
	\end{proposition}
	
	\section{Snark}
	The study of snark graphs were initiated in early 1880's from a classical problem in graph theory namely the Four-Colour theorem. The equivalence of Four-Colour	theorem with the fact that every bridgeless cubic graph is 3-colourable highlights the importance of the family of snark graphs.  The Petersen graph is considered as the smallest snark graph. In 1975 Isaacs found an infinite family of snarks namely Flower snarks \cite{flower}. The Szekeres snark was the other prominent one, discovered by George Szekeres in 1973 \cite{szwkers}. Further in 1976, F. Loupekine proposed a construction of two infinite families of snarks, using subgraphs of other known snarks. Although developed by Loupekine, this method was originally published by R. Isaacs \cite{loup}. In 1981, Goldberg added his contribution to the infinite family of snarks which is	named after him as Goldberg snark \cite{goldberg}.  Another well known snark on 50 vertices is the Watkins snark discovered by John J. Watkins in 1989 \cite{watkin}. The articles \cite{refgeneration,refflower,bla,refwatkin,refloupekine,refromansnark} provide further details on the definition and properties of the snark family. Driven by its physical importance and motivated by the interesting counter examples on snark graphs available in the literature, we have obtained the Italian domination number of some families of snark. 
    
\noindent To make this work easier to understand, let $f(v_1,v_2,v_3,\cdots,v_n)=(1,0,1,\cdots, 1)$ denotes the Italian dominating function $f$ assigns $1,0,1,\ldots,1$ to the vertices $v_1,v_2,$ $v_3,\cdots,v_n$ respectively.
	
	
	\subsection{ Flower Snarks}
	The flower snarks $F_n$ on $4n$ vertices, where $n$ is odd can be constructed with the following steps. First consider $n$ copies of the star graph on 4 vertices. Denote the central vertex of each star $A_i$ and the outer vertices as $B_i, C_i$ and $D_i$, resulting  a disconnected graph on $4n$ vertices with $3n$ edges $A_i  B_i, A_i  C_i$ and $A_i D_i$ for $1 \leq i \leq n $. Then construct the $n$-cycle $(B_1\cdots B_n)$ which adds $n$ edges. Finally construct the $2n$-cycle $(C_1... C_nD_1... D_n)$ which adds $2n$ edges. Figure \ref{flower} shows the flower snark $F_5$ constructed using 5 copies of star graph.
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		\caption{Flower snark $F_5$.}\label{flower}
	\end{figure}
	\begin{theorem}
		Let $F_n$ be a flower snark, where $n$ is odd. Then $\gamma_I(F_n$)= $2n-1$.
	\end{theorem}
	
	\begin{proof}
		Let $F_n$ be a flower snark with $4n$ vertices, where $n$ is odd, which can be constructed as described above. Consider an IDF, $f: V \rightarrow \{0, 1, 2\}$ defined as $f(B_1, B_2,B_3,\ldots, B_n)= (0, 1,   0,\ldots,0)$, $f(C_1,C_2,\ldots, C_n,D_1,D_2,\ldots ,D_n)=(0,1,0,\ldots, 0,0,1,0,\ldots ,0)$ and $f(A_1,A_2,A_3,\ldots, A_n)=(1,0,1,\ldots, 1)$. Then $w(f)= \lfloor\frac{n}{2}\rfloor+ 2  \lfloor\frac{n}{2}\rfloor + \lceil\frac{n}{2}\rceil = 2n-1$, $n$ is odd. This is the IDF that provides the minimal weight, because when considering the cycle of length $n$ by Theorem \ref{thmpc}, $\gamma_{I}(C_n) = \lceil \frac{n}{2}\rceil$, however here the IDF allocates 1 to just $\lfloor\frac{n}{2}\rfloor$ vertices. Furthermore, $F_n$ is a $3$-regular, with all vertices assigned 1 being adjacent to only the vertices assigned 0. Hence, $\gamma_I(F_n)= 2n-1$.
	\end{proof}
	
	
	\subsection{Goldberg Snarks}
	Goldberg snark  $G_n$ are recursive structures generated by the basic block graph $B_n$. The vertex and edge set of $B_n$ is defined as $V(B_n) = \{a_n, b_{n0}, b_{n1}, c_{n0}, c_{n1}, u_n,	v_n, w_n\}$, $E(B_n) = \{a_nv_n, v_nw_n, v_n u_n,$ $ u_n b_{n0}, b_{n0} b_{n1}, b_{n1} w_n,w_n c_{n0}, u_n c_{n1}, c_{n0} c_{n1}\}$. The graph in Figure \ref{gold}(a) is the basic block graph $B_n$.
	For $n$ odd, $n \geq 3$ graph $G_n$ is obtained from $n$ copies of $B_n$ by adding  a set of link edges $E_{nj}$, where $E_{nj} = \{c_{n1}c_{j0}, 
	b_{n1} b_{j0}, a_n a_j\}, j = n+1$.  The Goldberg snark $G_3$ shown in Figure \ref{gold}(b)  is obtained from adding link edges on basic 
	block graphs $B_1, B_2$ and $ B_3$. 
	
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				\node[label](15) at (-1.3,0) {$b_{10}$};
				\node[label](16) at (.3,-1) {$a_1$};
				\node[label](17) at (-1.3,2) {$c_{10}$};
				\node[label](18) at (1.3,2) {$c_{11}$};
				
				
				\node[vertex](21) at (3,-2) {};
				\node[vertex](22) at (3,-3) {};
				\node[vertex](23) at (3,-4) {};
				\node[vertex](24) at (2,-4) {};
				\node[vertex](25) at (2,-2) {};
				\node[vertex](26) at (1,-3) {};
				\node[vertex](27) at (4,-2) {};
				\node[vertex](28) at (4,-4) {};
				
				\node[label](41) at (3.3,-2) {$u_2$};
				\node[label](42) at (3.3,-3) {$v_2$};
				\node[label](43) at (3.3,-4) {$w_2$};
				\node[label](44) at (2,-4.2) {$b_{21}$};
				\node[label](45) at (2.2,-1.8) {$b_{20}$};
				\node[label](46) at (1.2,-2.8) {$a_2$};
				\node[label](47) at (4.3,-2) {$c_{20}$};
				\node[label](48) at (4.3,-4) {$c_{21}$};
				
				\node[vertex](31) at (-3,-2) {};
				\node[vertex](32) at (-3,-3) {};
				\node[vertex](33) at (-3,-4) {};
				\node[vertex](34) at (-2,-4) {};
				\node[vertex](35) at (-2,-2) {};
				\node[vertex](36) at (-1,-3) {};
				\node[vertex](37) at (-4,-2) {};
				\node[vertex](38) at (-4,-4) {};
				
				\node[label](51) at (-3.3,-2) {$w_3$};
				\node[label](52) at (-3.3,-3) {$v_3$};
				\node[label](53) at (-3.3,-4) {$u_3$};
				\node[label](54) at (-2,-4.2) {$b_{30}$};
				\node[label](55) at (-2.2,-1.8) {$b_{31}$};
				\node[label](56) at (-1.2,-2.8) {$a_3$};
				\node[label](57) at (-4.3,-2) {$c_{31}$};
				\node[label](58) at (-4.3,-4) {$c_{30}$};
				\node[label](59) at (-.1,-5.3) {(b)};
				
				
				
				
				\draw[edge](2)--(1);
				\draw[edge](2)--(3);
				\draw[edge](3)--(4);
				\draw[edge](5)--(4);
				\draw[edge](3)--(7);
				\draw[edge](6)--(2);
				\draw[edge](8)--(1);
				\draw[edge](5)--(1);
				\draw[edge](8)--(7);
				
				\draw[edge](22)--(21);
				\draw[edge](22)--(23);
				\draw[edge](23)--(24);
				\draw[edge](25)--(24);
				\draw[edge](23)--(27);
				\draw[edge](26)--(22);
				\draw[edge](28)--(21);
				\draw[edge](25)--(21);
				\draw[edge](28)--(27);
				
				\draw[edge](32)--(31);
				\draw[edge](32)--(33);
				\draw[edge](33)--(34);
				\draw[edge](35)--(34);
				\draw[edge](33)--(37);
				\draw[edge](36)--(32);
				\draw[edge](38)--(31);
				\draw[edge](35)--(31);
				\draw[edge](38)--(37);
				\draw[edge](6)--(26);
				\draw[edge](6)--(36);
				\draw[edge](36)--(26);
				\draw[edge](37)--(7);
				\draw[edge](8)--(27);
				\draw[edge](4)--(25);
				\draw[edge](5)--(35);
				\draw[edge](34)--(24);
				\draw (38) to[out=-20,in=-160] (28); 
		\end{tikzpicture}				}
		\caption{(a) Basic block $B_n$. (b) Goldberg snark $G_3$ constructed from $B_1, B_2$ and $ B_3$.}\label{gold}
	\end{figure}
	
	\begin{theorem}
		Let $G_n$ be a Goldberg snark, where $n$  is odd. Then $\gamma_I(G_n)=3n +\lfloor\frac{n}{2}\rfloor$.
	\end{theorem}
	\begin{proof}
		Let $G_n$ be a Goldberg snark and $f$ be an IDF on $V(G_n)$. In the graph, $B=\{b_{10}, b_{11}, b_{20}, b_{21}, \ldots, b_{n0}, b_{n1} \}$ and $C=\{c_{10}, c_{11}, c_{20}, c_{21},$  $ \ldots, c_{n0}, c_{n1}\}$ forms a $2n$-cycle. So by the Theorem \ref{thmpc}, $f$ can assign 0 and 1 alternatively to the vertices in $B$ and $C$. In the following step, $f(u_i)=0$, $f(w_i)=0$, and $f(v_i)=1$, for $1\leq i\leq n$. Note that so far all the vertices labeled 1 are adjacent to only the vertices labeled 0, through which it is not possible to reduce the weight $w(f)=3n$ further. To meet the IDF condition, we must label the remaining vertices $A=\{a_1, a_2, \ldots, a_n\}$ that form a $n$-cycle with 0 and 1 in alternate order.  Thus the Italian dominating function defined above is the function that offers the least weight because each vertex labeled 1 is adjacent to only at most one vertex labeled 1. Hence  $\gamma_I(G_n)= 3n+\lfloor\frac{n}{2}\rfloor$.
	\end{proof}
	
	
	
	
	
	
	\subsection{Loupekine snarks}
	$LP_1$-snarks are an infinite family of Loupekine snarks whose construction is presented below. Let $k$ be an odd positive integer. A $k$-$LP_1$-snark  is
	constructed from $k \geq 3$ sub graphs called blocks $B_i$ for $1\leq i\leq k$, obtained from the Petersen graph. Figure \ref{loup} (a) illustrates an arbitrary block $B_i$ with its vertices named $x_i, u_i, w_i, v_i, y_i$, commonly called border vertices.  For all $i\in \{1,\ldots,k\}$, the border vertices $v_i$ and $y_i$ of block $B_i$ are linked to the border vertices $u_{i+1}$ and $x_{i+1}$ of block $B_{i+1}$ (indices taken modulo k) by edges called linking edges. The linking edges can be $\{v_ix_{i+1}, y_{i}u_{i+1}\}$ or $\{v_iu_{i+1},y_{i}x_{i+1}\}$, but not both. 
	The resulting graph, denoted by $G_B$, is the block subgraph of $k$-$LP_1$-snark. Figure \ref{loup 2} with out dotted edges shows an example of $G_B$ constructed using the linking edges  $\{v_iu_{i+1},y_{i}x_{i+1}\}$.\\
	Notice that $G_B$ has exactly $k$ vertices with degree two $:w_1,\ldots w_k$. Let $G_C$ be a central subgraph of $k$-$LP_1$-snark  with exactly $k$ vertices of degree one, namely $z_1,\cdots,z_k$, and all the other vertices of degree three such that each connected component of $G_C$ is isomorphic to one of the graphs $K_2$ and $S_3$, with $K_2$ being
	a complete graph with two vertices, and $S_3$ a star with three vertices of degree one (see Figure \ref{loup} (b)).   For each $i$, with $1 \leq i \leq k$,
	identify vertices $w_i$ and $z_i$, thus concluding the construction of a $k$-$LP_1$-snark. 
	
	\begin{figure}[h]
		\centering
		{%
			\begin{tikzpicture}
				\tikzstyle{vertex} = [circle,fill=black,draw,minimum size=.15cm,inner sep=0pt]
				\tikzstyle{label} = []
				\tikzstyle{edge} = [-, thin]
				\
				\node [vertex](1) at (0,-0.7) {};
				\node [vertex](2) at (-1.4,-1.5) {};
				\node [vertex](3) at (1.4,-1.5) {};
				\node [vertex](4) at (-1,-3) {};
				\node [vertex](5) at (1,-3) {};
				\node [vertex](6) at (-1,-3.5) {};
				\node [vertex](7) at (1,-3.5) {};
				
				\node [vertex](15) at (8,-.9) {};
				\node [vertex](16) at (8,-2) {};
				\node [vertex](17) at (7,-1.5) {};
				\node [vertex](18) at (7,-3.5) {};
				\node [vertex](19) at (9,-1.5) {};
				\node [vertex](20) at (9,-3.5) {};
				\node [vertex](21) at (6,-3.2) {};
				\node [vertex](22) at (10, -3.2) {};
				\node[label](8) at (0.3,-.7) {$w_i$};
				\node[label](9) at (-1.7,-1.5) {$v_i$};
				\node[label](10) at (1.7,-1.5) {$u_i$};
				\node[label](11) at (-1.3,-3) {$t_i$};
				\node[label](12) at (1.3,-3) {$r_i$};
				\node[label](13) at (-1.3,-3.5) {$y_i$};
				\node[label](14) at (1.3,-3.5) {$x_i$};
				\node[label](23) at (0,-4) {$(a)$};
				\node[label](24) at (8,-4) {$(b)$};
				\draw[edge](1)--(4);
				\draw[edge](1)--(5);
				\draw[edge](2)--(3);
				\draw[edge](2)--(5);
				\draw[edge](3)--(4);
				
				\draw[edge](4)--(6);
				\draw[edge](5)--(7);
				\draw[edge](6)--(7);
				\draw[edge] (17) to[bend left](18);
				\draw[edge] (19) to[bend right](20);
				\draw[edge](15)--(16);
				\draw[edge](16)--(21);
				\draw[edge](16)--(22);
			\end{tikzpicture}		
		}
		\caption{(a) Basic Block $B_i$  (b) A graph $G_c$ with seven pendant vertices.}\label{loup}
	\end{figure}
	
	\begin{figure}[h]
		\centering
		%\resizebox{5cm}{5cm}
		{%
			\begin{tikzpicture}
				\tikzstyle{vertex} = [circle,fill=black,draw,minimum size=.15cm,inner sep=0pt]
				\tikzstyle{label} = []
				
				\tikzstyle{edge} = [-]
				\tikzstyle{ed} = [-,very thick, dotted]
				\tikzstyle{edg} = [-,very thick, dotted]
				\tikzstyle{ed} = [-,very thick, dotted]
				\node[vertex](1) at (-.7,1) {};
				\node[vertex](3) at (.7,1) {};
				\node[vertex](4) at (1,0) {};
				\node[vertex](5) at (-1,0) {};
				\node[vertex](6) at (0,-.8) {};
				\node[vertex](7) at (-.7,2) {};
				\node[vertex](8) at (.7,2) {};
				\node[vertex](9) at (0,-2.5) {};
				
				
				
				\node[label](11) at (-1,1) {$r_1$};
				\node[label](13) at (1,1) {$t_1$};
				\node[label](14) at (1.3,0) {$v_{1}$};
				\node[label](15) at (-1.3,0) {$u_{1}$};
				\node[label](16) at (.3,-.8) {$w_1$};
				\node[label](17) at (-1,2) {$x_1$};
				\node[label](18) at (1,2) {$y_1$};
				
				\node[vertex](21) at (4,-2.4) {};
				\node[vertex](23) at (4,-3.6) {};
				\node[vertex](24) at (3,-4) {};
				\node[vertex](25) at (3,-2) {};
				\node[vertex](26) at (2.2,-3) {};
				\node[vertex](27) at (5,-2.4) {};
				\node[vertex](28) at (5,-3.6) {};
				
				\node[label](41) at (4.2,-2.2) {$r_2$};
				\node[label](43) at (4.2,-3.8) {$t_2$};
				\node[label](44) at (3,-4.2) {$v_{2}$};
				\node[label](45) at (3.2,-1.8) {$u_{2}$};
				\node[label](46) at (2.2,-2.8) {$w_2$};
				\node[label](47) at (5.2,-3.8) {$y_2$};
				\node[label](48) at (5.2,-2.2) {$x_2$};
				
				
				\node[vertex](31) at (-4,-2.4) {};
				\node[vertex](33) at (-4,-3.6) {};
				\node[vertex](34) at (-3,-4) {};
				\node[vertex](35) at (-3,-2) {};
				\node[vertex](36) at (-2.2,-3) {};
				\node[vertex](37) at (-5,-2.4) {};
				\node[vertex](38) at (-5,-3.6) {};
				
				\node[label](51) at (-4.3,-2.2) {$t_3$};
				\node[label](53) at (-4.3,-3.8) {$r_3$};
				\node[label](54) at (-3,-4.2) {$u_{3}$};
				\node[label](55) at (-3.2,-1.8) {$v_{3}$};
				\node[label](56) at (-2.2,-2.8) {$w_3$};
				\node[label](57) at (-5.3,-2.2) {$y_3$};
				\node[label](58) at (-5.3,-3.8) {$x_3$};
				
				
				\draw[edge](1)--(7);
				\draw[edge](3)--(8);
				\draw[edge](21)--(27);
				\draw[edge](23)--(28);
				\draw[edge](31)--(37);
				\draw[edge](33)--(38);
				\draw[edge](7)--(8);
				\draw[edge](27)--(28);
				\draw[edge](37)--(38);
				\draw[edge](37)--(7);
				\draw (38) to[out=-30,in=-150] (28); 
				\draw[edge](8)--(27);
				\draw[edge](5)--(35);
				\draw[edge](4)--(25);
				\draw[edge](34)--(24);
				
				\draw[edg](9)--(6);
				\draw[edg](9)--(26);
				\draw[edg](9)--(36);
				
				\draw[edge](5)--(4);
				\draw[edge](5)--(3);
				\draw[edge](3)--(6);
				\draw[edge](4)--(1);
				\draw[edge](6)--(1);
				\draw[edge](25)--(24);
				\draw[edge](25)--(23);
				\draw[edge](23)--(26);
				\draw[edge](24)--(21);
				\draw[edge](26)--(21);
				\draw[edge](5)--(4);
				\draw[edge](35)--(34);
				\draw[edge](35)--(33);
				\draw[edge](33)--(36);
				\draw[edge](34)--(31);
				\draw[edge](36)--(31);
				
				
		\end{tikzpicture}				}
		\caption{Loupekine snark $3$-$LP_1$.}\label{loup 2}
	\end{figure}
	
	
	\begin{theorem}
		Let $k$-$LP_1$ be a Loupekine Snark for $k\geq 3 $ and $k$ is odd. Then $\gamma_I(k$-$LP_1)=3k$.
	\end{theorem} 
	\begin{proof}
		Let $k$-$LP_1$ be a Loupekine Snark with $7k+1$ vertices, developed from subgraph $G_B$ with linking edges $\{v_iu_{i+1},y_{i}x_{i+1}\}$ and central graph $G_C$ as defined in the construction.
		Consider the subgraph $G_B$ and an Italian dominating function $f$ on $V(G_B)$. Since $X=\{x_1,y_1,x_2,y_2,\ldots,x_k,y_k\}$ and $U=\{u_1,v_1,u_2,v_2,\ldots,u_k,v_k\}$ form a two disjoint $2k$ cycle, by Theorem \ref{thmpc}, $f$ can assign 0 and 1 alternatively to the vertices both in $X$ and $Y$. For all $1 \leq i \leq k$, both vertices $r_i$ and $t_i$ are adjacent to $w_i$, and just one vertex assigned 1 in cycles. When $f$ assigns 1 to $w_i$ and 0 to $r_i$ and $t_i$, for all $1 \leq i \leq k$, all vertices assigned $0$ are adjacent to exactly two vertices assigned $1$. Hence, the total weight $w(f)=3k$ is minimal for the subgraph $G_B$. Finally to form  $k$-$LP_1$ Loupekine Snark the vertex $w_i$ is identified with $z_i$, for all $1 \leq i \leq k$. Then the IDF $f$ defined above can be extended to $k$-$LP_1$ Loupekine Snark, by assigning $0$ to the central vertices of star mentioned in $G_c$, since as $k$ is odd, at least one set of any three $w_i$ is adjacent to the central vertex of star. Hence minimum weight of an Italian dominating function on $k$-$LP_1$ Loupekine Snark is $3k$, where $k\geq 3$ and $k$ is odd.
	\end{proof}
	\subsection{Watkin Snark}
	Watkins snark  is a snark on 50 vertices and 75 edges constructed using the 5 copies of basic block shown in Figure \ref{watkin block}. Watkin snark  constructed from basic block graph $B_i$, for $1 \leq i \leq 5$ is illustrated in Figure \ref{watkin}.
	\begin{figure}[h]
		\centering
		%\resizebox{5cm}{5cm}
		{%
			\begin{tikzpicture}
				\tikzstyle{vertex} = [circle,fill=black,draw,minimum size=.15cm,inner sep=0pt]
				\tikzstyle{label} = []
				%\tikzstyle{label} = [circle,fill=white]
				%\tikzstyle{label} = [fill=white]
				\tikzstyle{edge} = [-]
				\tikzstyle{ed} = [-,very thick, dotted]
				\tikzstyle{edg} = [-,very thick]
				\tikzstyle{ed} = [-,very thick, dotted]
				
				
				\node[vertex](21) at (5.5,1.3) {};
				\node[vertex](22) at (6.5,.6) {};
				\node[vertex](23) at (7.5,1.3) {};
				\node[vertex](24) at (7.5,0.6) {};
				\node[vertex](25) at (5.5,0.6) {};
				\node[vertex](26) at (5.5,-.3) {};
				\node[vertex](210) at (7.5,-.3) {};
				\node[vertex](27) at (5,2) {};
				\node[vertex](28) at (8,2) {};
				\node[vertex](29) at (6.5,2) {};
				
				
				\node[label](31) at (5.2,1.3) {0};
				\node[label](32) at (6.5,0.3) {0};
				\node[label](33) at (7.8,1.3) {1};
				\node[label](34) at (7.8,0.6) {0};
				\node[label](35) at (5.2,0.6) {1};
				\node[label](36) at (5.2,-.2) {0};
				\node[label](310) at (7.8,-.2) {1};
				\node[label](37) at (5,2.3) {0};
				\node[label](38) at (8,2.3) {0};
				\node[label](39) at (6.5,2.3) {1};
				
				\draw[edge](23)--(21);
				\draw[edge](22)--(29);
				\draw[edge](23)--(24);
				\draw[edge](25)--(24);
				\draw[edge](23)--(28);
				\draw[edge](26)--(25);
				\draw[edge](24)--(210);
				\draw[edge](26)--(210);
				\draw[edge](27)--(21);
				\draw[edge](25)--(21);
				\draw[edge](28)--(27);
				
		\end{tikzpicture}				}
		\caption{An IDF assigned on basic block of Watkin.}\label{watkin block}
	\end{figure}
	
	\begin{figure}[h!]
		\centering
		%\resizebox{7cm}{7cm}
		{%
			\begin{tikzpicture}
				\tikzstyle{vertex} = [circle,fill=black,draw,minimum size=.15cm,inner sep=0pt]
				\tikzstyle{label} = []
				\tikzstyle{edge} = [-]
				\node [vertex](1) at (0,3) {};
				\node [vertex](2) at (1,3) {};
				\node [vertex](3) at (-1,3) {};
				\node [vertex](4) at (1,4) {};
				\node [vertex](5) at (-1,4) {};
				\node [vertex](6) at (0,5) {};
				\node [vertex](7) at (-2,5) {};
				\node [vertex](8) at (2,5) {};
				\node [vertex](9) at (-1,2) {};
				\node [vertex](10) at (1,2) {};
				
				\node [vertex](11) at (6,2.5) {};
				\node [vertex](12) at (6,0.5) {};
				\node [vertex](13) at (6,-1.5) {};
				\node [vertex](14) at (5,1.5) {};
				\node [vertex](15) at (5,-0.5) {};
				\node [vertex](16) at (4,1.5) {};
				\node [vertex](17) at (4,-.5) {};
				\node [vertex](18) at (4,.5) {};
				\node [vertex](19) at (3,1.5) {};
				\node [vertex](20) at (3,-.5) {};
				
				\node [vertex](21) at (-6,2.5) {};
				\node [vertex](22) at (-6,0.5) {};
				\node [vertex](23) at (-6,-1.5) {};
				\node [vertex](24) at (-5,1.5) {};
				\node [vertex](25) at (-5,-.5) {};
				\node [vertex](26) at (-4,1.5) {};
				\node [vertex](27) at (-4,-.5) {};
				\node [vertex](28) at (-4,.5) {};
				\node [vertex](29) at (-3,1.5) {};
				\node [vertex](30) at (-3,-.5) {};
				
				\node [vertex](31) at (2.5,-1.9) {};
				\node [vertex](32) at (1,-2.8) {};
				\node [vertex](33) at (3.2,-2.8) {};
				\node [vertex](34) at (1.6,-3.8) {};
				\node [vertex](35) at (2.4,-3.3) {};
				\node [vertex](36) at (3.8,-3.6) {};
				\node [vertex](37) at (2,-4.7) {};
				\node [vertex](38) at (4.9, -4.2) {};
				\node [vertex](39) at (2,-6) {};
				\node [vertex](40) at (3.4,-5.1) {};
				
				\node [vertex](41) at (-2.5,-1.9) {};
				\node [vertex](42) at (-1,-2.8) {};
				\node [vertex](43) at (-3.2,-2.8) {};
				\node [vertex](44) at (-1.6,-3.8) {};
				\node [vertex](45) at (-2.4,-3.3) {};
				\node [vertex](46) at (-3.8,-3.6) {};
				\node [vertex](47) at (-2,-4.7) {};
				\node [vertex](48) at (-4.9, -4.2) {};
				\node [vertex](49) at (-2,-6) {};
				\node [vertex](50) at (-3.4,-5.1) {};
				
				
				
				\node [label](91) at (0,2.7) {0};
				\node [label](92) at (1.2,3) {0};
				\node [label](93) at (-1.2,3) {1};
				\node [label](94) at (1,4.3) {1};
				\node [label](95) at (-1,4.3) {0};
				\node [label](96) at (0,5.3) {1};
				\node [label](97) at (-2,5.3) {0};
				\node [label](98) at (2,5.3) {0};
				\node [label](99) at (-1.2,2.2) {0};
				\node [label](910) at (1.2,2.2) {1};
				
				\node [label](911) at (6.3,2.5) {0};
				\node [label](912) at (6.3,0.5) {1};
				\node [label](913) at (6.3,-1.5) {0};
				\node [label](914) at (5,1.8) {0};
				\node [label](915) at (5,-0.8) {1};
				\node [label](916) at (4,1.8) {1};
				\node [label](917) at (4,-.8) {0};
				\node [label](918) at (3.7,.5) {0};
				\node [label](919) at (3.1,1.8) {0};
				\node [label](920) at (3,-.8) {1};
				
				\node [label](921) at (-6.3,2.5) {0};
				\node [label](922) at (-6.3,0.5) {1};
				\node [label](923) at (-6.3,-1.5) {0};
				\node [label](924) at (-5,1.8) {1};
				\node [label](925) at (-5,-.8) {0};
				\node [label](926) at (-4,1.8) {0};
				\node [label](927) at (-4,-.8) {1};
				\node [label](928) at (-3.7,.5) {0};
				\node [label](929) at (-3,1.8) {1};
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				\draw[edge](40)--(35);
				
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				\draw[edge](44)--(47);
				\draw[edge](46)--(47);
				\draw[edge](46)--(48);
				\draw[edge](47)--(49);
				\draw[edge](48)--(50);
				\draw[edge](49)--(50);
				\draw[edge](50)--(45);
				
				\draw[edge](8)--(19);
				
				\draw[edge](10)--(38);
				\draw[edge](13)--(31);
				\draw[edge](20)--(49);
				\draw[edge](32)--(23);
				\draw[edge](39)--(42);
				\draw[edge](48)--(30);
				\draw[edge](41)--(7);
				\draw[edge](21)--(9);
				\draw[edge](29)--(11);
			\end{tikzpicture}		
		}
		\caption{Watkins Snark}\label{watkin}
	\end{figure}
	\begin{theorem}
		Let $W$ be the Watkin snark. Then $\gamma_{I}(W)=20$.
	\end{theorem}
	\begin{proof}
		Let $W$ be the Watkin snark with 50 vertices and $B$ be the basic block used to construct the Watkin snark. Let $f$ be an Italian domination function assigned for the vertices of each block $B_i$, for $1 \leq i \leq 5$, as shown in Figure \ref{watkin block}. While examining each block, the IDF condition fails for only one two-degree vertex labeled 0. However, they are connected to the vertex assigned 1 during the formation of Watkin snark (see Figure \ref{watkin}). Therefore, $\gamma_{I}(W)\leq 5w(f)=20$. According to Theorem \ref{lb}, $\gamma_{I}(G) \geq \frac{2n}{\Delta + 2}=\frac{100}{5}=20$. Hence $\gamma_{I}(G) =20$.
	\end{proof}
	\subsection{Szekeres Snark} 
	Szekeres snark is a snark with 50 vertices and 75 edges. Figure \ref{szekeres} shows an IDF assigned on the Szekeres graph.
	\begin{figure}[h!]
		\centering
		%	\resizebox{7cm}{7cm}
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				\node[label](51) at (0,4.7) {$0$};
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				\node[label](59) at (1.65,2) {$1$};
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				\node[label](77) at (-3.8,-0.75){$0$};
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				\draw[edge](41)--(42);
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				\draw[edge](44)--(49);
				\draw[edge](48)--(49);
				\draw[edge](47)--(48);
				\draw[edge](46)--(47);
				\draw[edge](45)--(46);
				\draw[edge](41)--(49);
				\draw[edge](44)--(50);
				\draw[edge](42)--(50);
				\draw[edge](47)--(50);
				
				
				\draw[edge](6)--(33);
				\draw[edge](4)--(43);
				\draw[edge](15)--(25);
				\draw[edge](31)--(41);
				\draw[edge](11)--(36);
				\draw[edge](21)--(46);
				\draw[edge](9)--(12);
				\draw[edge](10)--(22);
				\draw[edge](17)--(48);
				\draw[edge](27)--(38);
				
			\end{tikzpicture}	
		}
		\caption{Szekeres Snark}\label{szekeres}
	\end{figure}
	\begin{theorem}
		Let $G$ be the Szekeres snark. Then $\gamma_{I}(G)=20$.
	\end{theorem}
	\begin{proof}
		Let $G$ be the Szekeres snark with 50 vertices and $f$ be the IDF for Snekers snark as assigned in Figure \ref{szekeres}. Then $\gamma_{I}(G)\leq w(f)=20$. According to Theorem \ref{lb}, $\gamma_{I}(G)\geq \frac{2n}{\Delta+2}\geq 20$. Hence $\gamma_{I}(G)=20$.
	\end{proof}
	
\section{Conclusion and future scope}
	In this paper we found the exact values of Italian domination number of some families of snark. In the near future, one can explore the Italian domination number of other classes of graphs.
	

 
\bigskip
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\end{thebibliography}
\end{document}
