Abstract
A complete secure dominating set of a graph $G$ is a dominating set $D \subseteq V(G)$ with the property that for each $v \in D$, there exists $F=\lbrace v_{j} \vert v_{j} \in N(v) \cap (V(G)-D)\rbrace$, such that for each $v_{j} \in F$, $( D-\lbrace v \rbrace) \cup \lbrace v_{j} \rbrace$ is a dominating set. The minimum cardinality of any complete secure dominating set is called the complete secure domination number of $G$ and is denoted by $\gamma_{csd}(G)$. In this paper, the bounds for complete secure domination number for some standard graphs like grid graphs and stacked prism graphs in terms of number of vertices of $G$ are found and also the bounds for the complete secure domination number of a tree are obtained in terms of different parameters of $G$.
Highlights
The graphs considered here are undirected, finite, connected, without multiple edges or loops and without isolated vertices
The bounds for complete secure domination number for some standard graphs like grid graphs and stacked prism graphs in terms of number of vertices of G are found and the bounds for the complete secure domination number of a tree are obtained in terms of different parameters of G
A secure dominating set X of a graph G is a dominating set with the property that each vertex u ∈ V (G) − X is adjacent to a vertex v ∈ X such that(X − {v}) ∪ {u} is dominating set
Summary
The graphs considered here are undirected, finite, connected, without multiple edges or loops and without isolated vertices. A set of vertices D is said to dominate the graph G if for each vertex v ∈ V (G) − D , there is a vertex u ∈ D with v is adjacent to u. The minimum cardinality of any dominating set is called the domination number of G and it is denoted by γ(G). The minimum cardinality of such a set is called the secure domination number, denoted by γs(G). The minimum cardinality of any complete secure dominating set is called the complete secure domination number of G and is denoted by γcsd(G)
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