Sep 2026· Boletim da Sociedade Paranaense de Matemática· Vol 44, pp. 1-6· 0 citations· 12 references
Abstract
Let $G$ be a graph of minimum degree at least two. A set $D \subseteq V(G)$ is said to be a
\emph{double total dominating set} of $G$, for every vertex $v\in V (G) $ contains at least two neighbors in $D$. The minimum cardinality among all double total dominating sets of $G$ is the \emph{double total
domination number} (DTDN) of $G$, denoted by $\gamma_{2t}(G)$. In this paper, we obtain the DTDN of wheel graphs $W_n$ as $1+\gamma_{t}(W_{n})$, where $\gamma_t(G)$ be the total domination number of $G$. Also, we obtain the DTDN of generalized wheel graphs $M_n$, namely, $\gamma_{2t}(M_{n})= 2\left\lceil\frac{n}{3}\right\rceil$ if $n$ is even and $\gamma_{2t}(M_{n})= \frac{n+1}{2}$ if $n$ is odd. Further, we find the DTDN of Myceilskian of any graph $G$ is either $\gamma_{2t}(G)+1$ or $\gamma_{2t}(G)+2$. We find the DTDN of Myceilskian of path $P_n$ is $n+2$ if $n$ is odd and $4\left\lfloor\frac{n-2}{4}\right\rfloor+5$ if $n$ is even. Also, we find the DTDN of Myceilskian of cycle $C_n$, complete graph and complete $m$-partite graph are $n+1$, $5$, $5$ and $5$ respectively.
A set $D\subseteq V(G)$ is a $1$-limited dominating set of a graph $G$ if every vertex outside $D$ has a neighbor in $D$ and every vertex of $D$ has at most one neighbor outside $D$. The minimum cardinality among all $1$-limited dominating sets of $G$ is denoted by $\gamma_1^{\mathrm L}(G)$. For every cubic graph $G$ o...
Goran Radić, Gordana Radić, Aleksandra Tepeh· 0 citations
The total domination number $\gamma_t(G)$ of a graph $G$ is the minimum cardinality of a set $D\subseteq V(G)$ such that every vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ is the largest integer $k$ for which the sum of the $k$ smallest degrees of $G$ is at most $|E(G)|$. A well-known conjecture,...
A dominating set $D$ of a graph $G$ is a \emph{fair dominating set} if every two vertices outside $D$ have the same number of neighbors in $D$, and the \emph{fair domination number} $\mathrm{fd}(G)$ is the minimum cardinality of such a set. Caro, Hansberg and Henning, who introduced this parameter, proved that $\mathrm...
A paired dominating set of a graph $G$ is a dominating set $D$ such that $G[D]$ has a perfect matching. The minimum size of such a set is the paired domination number $\gpr(G)$. Desormeaux and Henning conjectured that every cubic bipartite graph $G$ of order $n$ satisfies $\gpr(G)\le n/2$. We prove the conjecture in th...
An edge labeling of a connected graph $G=(V, E)$ is called a local antimagic labeling if it is a bijection $f: E\rightarrow \{1, 2, \cdots, \left | E \right | \}$ such that for any pair of adjacent vertices $u$ and $v$, $f^{+}(u)\neq f^{+}(v)$ where $f^{+}(x)=\sum f(e)$, with $e$ ranging over all the edges incident to...
For a graph $G$ and a set $B\subseteq V(G)$, the spread $\mathrm{sp}(B)$ of $B$ is the difference between the largest and the smallest degree in $G$ of a vertex of $B$, and for an integer $k\geq 0$ the parameter $\mathrm{sp}(G,k)$ is the largest cardinality of a set $B$ with $\mathrm{sp}(B)\leq k$. Caro, Lauri and Zarb...
Y. Caro, R. Škrekovski, Christina Zarb· 0 citations
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