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Domination, total domination, and component domination
Discussiones Mathematicae Graph TheoryTeresa Haynes, Michael A. Henning
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On the total domination number of total graphs
Discussiones Mathematicae Graph TheorySummary: Let \(G\) be a graph with no isolated vertex. A set \(D\subseteq V(G)\) is a total dominating set of \(G\) if every vertex of \(G\) is adjacent to at least one vertex in \(D\). The total domination number of \(G\), denoted by \(\gamma_t(G)\), is the minimum cardinality among all total dominating sets of \(G\).
Abel Cabrera-Martínez +2 more
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Strong total domination and weak total domination in Mycielski’s graphs
Malaya Journal of Matematik, 2020 Let \(G=(V, E)\) be a graph. A set \(S \subseteq V\) is called a weak total dominating set (WTD-set) if each vertex \(v \in V-S\) is adjacent to a vertex \(u \in S\) with \(\operatorname{deg}(v)>\operatorname{deg}(u)\) and every vertex in \(S\) adjacent to a vertex in \(S\).
TUNÇEL GÖLPEK, HANDE, AYTAÇ, AYSUN
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On upper bounds for total $k$-domination number via the probabilistic method [PDF]
, 2023 summary:For a fixed positive integer $k$ and $G=(V, E)$ a connected graph of order $n$, whose minimum vertex degree is at least $k$, a set $S\subseteq V$ is a total $k$-dominating set, also known as a $k$-tuple total dominating set, if every vertex $v\in
Cruz-Suárez, Hugo +2 more
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On graphs with equal total domination and Grundy total domination numbers
Aequationes mathematicae, 2021 A sequence $(v_1,\ldots ,v_k)$ of vertices in a graph $G$ without isolated vertices is called a total dominating sequence if every vertex $v_i$ in the sequence totally dominates at least one vertex that was not totally dominated by $\{v_1,\ldots , v_{i-1}\}$ and $\{v_1,\ldots ,v_k\}$ is a total dominating set of $G$.
Tanja Dravec +3 more
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Total dominator total coloring of a graph
Contributions to Discrete Mathematics, 2023 Here, we initiate to study the total dominator total coloring of a graph which is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. In more details, while in section 2 we present some tight lower and upper bounds for the total dominator total chromatic number of a graphs in ...
Adel P. Kazemi +2 more
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Singed Total Domatic Number of a Graph [PDF]
, 2010 The maximum number of functions in a signed total dominating family on G is the signed total domatic number of G. In this paper, some properties related signed total domatic number and signed total domination number of a graph are studied and found the ...
Shailaja S. Shirkol +2 more
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The Electronic Journal of Combinatorics, 2017 Using hypergraph transversals it is proved that $\gamma_t(Q_{n+1}) = 2\gamma(Q_n)$, where $\gamma_t(G)$ and $\gamma(G)$ denote the total domination number and the domination number of $G$, respectively, and $Q_n$ is the $n$-dimensional hypercube. More generally, it is shown that if $G$ is a bipartite graph, then $\gamma_t(G \square K_2) = 2\gamma(G ...
Jernej Azarija +2 more
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An upper bound on the total outer-independent domination number of a tree [PDF]
Opuscula Mathematica, 2012 A total outer-independent dominating set of a graph \(G=(V(G),E(G))\) is a set \(D\) of vertices of \(G\) such that every vertex of \(G\) has a neighbor in \(D\), and the set \(V(G) \setminus D\) is independent.
Marcin Krzywkowski
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Perfect Domination Excellent Trees [PDF]
, 2012 A set D of vertices of a graph G is a perfect dominating set if every vertex in V \ D is adjacent to exactly one vertex in D. In this paper we introduce the concept of perfect domination excellent graph as a graph in which every vertex belongs to some ...
Sharada, B., Sharada B.
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