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A dominating set D ⊂ V (G) is said to be a captive dominating set of G if 〈 D 〉 has no isolated vertex (D is a total dominating set) and each vertex v ∈ D is adjacent to at least one vertex in V − D. A captive dominating set D is said to be a minimal captive dominating set if no proper subset D ′ of D is a captive dominating set. The minimum cardinality of a minimal captive dominating set of G is called the captive domination number of G which is denoted by γ c a (G). In this paper, we have characterized some results, determine the values of the domination-related parameters for the graph and its splitting graph, relate captive domination and packing number of a graph, etc. We have also constructed graphs for which γ c a > γ t. [ABSTRACT FROM AUTHOR]

A set S ⊆ V (G) of vertices in a graph G is called a packing of G if the closed neighborhood of the vertices of S are pairwise disjoint in G. A subset S of V (G) is called an open packing of G if the open neighborhood of the vertices of S are pairwise disjoint in G. We have investigated exact value of these parameters for triangular snakes. [ABSTRACT FROM AUTHOR]

The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γ(G-E0) > γ (G). An equitable dominating set D is called a total equitable dominating set if the induced subgraph < D > has no isolated vertices. The total equitable domination number γte(G) of G is the minimum cardinality of a total equitable dominating set of G. If γte(G) ≠ |V(G)| and contains no isolated vertices then the total equitable bondage number bte(G) of a graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γte(G-E0) > γte(G). In the present work we prove some characterizations and investigate total equitable bondage number of Ladder and degree splitting of path. [ABSTRACT FROM AUTHOR]

In this paper we determine distance k--domination number of graph obtained by duplication of vertices altogether by edges in cycle Cn, splitting graph of cycle Cn as well as graph obtained by duplication of edges altogether by vertices in cycle Cn. [ABSTRACT FROM AUTHOR]