Inverse and Disjoint Secure Total Domination in Graphs
نویسنده
چکیده
Let G = (V, E) be a graph. Let D be a minimum secure total dominating set of G. If V – D contains a secure total dominating set D' of G, then D' is called an inverse secure total dominating set with respect to D. The inverse secure total domination number γst(G) of G is the minimum cardinality of an inverse secure total dominating set of G. The disjoint secure total domination number γstγst(G) of G is the minimum cardinality of the union of two disjoint secure total dominating sets in G. In this paper, we initiate a study of these two parameters and establish some results on these new parameters.
منابع مشابه
Inverse and Disjoint Restrained Domination in Graphs
Let D be a minimum secure restrained dominating set of a graph G = (V, E). If V – D contains a restrained dominating set D' of G, then D' is called an inverse restrained dominating set with respect to D. The inverse restrained domination number γr(G) of G is the minimum cardinality of an inverse restrained dominating set of G. The disjoint restrained domination number γrγr(G) of G is the minimu...
متن کاملTOTAL DOMINATION POLYNOMIAL OF GRAPHS FROM PRIMARY SUBGRAPHS
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=sum d_t(G,i)$, where $d_t(G,i)$ is the numbe...
متن کاملA Characterization of Graphs with Disjoint Total Dominating Sets
A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. A fundamental problem in total domination theory in graphs is to determine which graphs have two disjoint total dominating sets. In this paper, we solve this problem and provide a constructive characterization of the graphs that have two disjoint total dominating sets.
متن کاملSigned total Italian k-domination in graphs
Let k ≥ 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function (STIkDF) on a graph G is a functionf : V (G) → {−1, 1, 2} satisfying the conditions that $sum_{xin N(v)}f(x)ge k$ for each vertex v ∈ V (G), where N(v) is the neighborhood of $v$, and each vertex u with f(u)=-1 is adjacent to a vertex v with f(v)=2 or to two vertic...
متن کاملTotal domination in $K_r$-covered graphs
The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...
متن کامل