For notation and graph theory terminology, we in general follow [1,2]. Specifically, let be a graph with vertex set of order and size , and let be a vertex in . The open neighborhood of is and the closed neighborhood of is . The degree of is . If the graph is clear from the context, we simply write and rather than and , respectively. For a set , its open neighborhood is the set , and its closed neighborhood is the set . A vertex of degree one is called a leaf and its unique neighbor a support vertex. A pendant edge is an edge which one of its end-points is a leaf. A star of order is a tree that has precisely one vertex that is not a leaf. A claw-free graph is a graph with no induced subgraph isomorphic to a star of order . A double-star is a tree that has precisely two vertices that are not leaves. We refer as a double-star which its central vertices have degree and , respectively. For a subset of vertices of we denote by the subgraph of induced by . For two subsets of vertices and of , we denote by the subgraph of induced by . The diameter of a graph , denoted by , is the maximum distance between pairs of vertices of . The girth of , denoted by , is the length of a shortest T-5224 contained in .
A subset of vertices of a graph is a dominating set of if every vertex in has a neighbor in . The domination number is the minimum cardinality of a dominating set of . We refer a dominating set of cardinality as a -set. A dominating set in a graph with no isolated vertex is called a total dominating set of if has no isolated vertex. The total domination number is the minimum cardinality of a total dominating set of . We refer a total dominating set of cardinality as a -set. A total dominating set in is called an efficient total dominating set if the open neighborhoods of the vertices of form a partition for . For a subset of vertices of , and a vertex , we say that a vertex is an external private neighbor ofwith respect to if . We denote by the set of all external private neighbors of respect to .
Hamid and Balamurugan  initiated the study of isolate domination in graphs. A dominating set is an isolate dominating set if the induced subgraph has at least one isolated vertex. The isolate domination number is the minimum cardinality of an isolate dominating set of . The concept of isolate domination was further studied, for example in [4–9]. Hamid et al.  showed that for a cubic graph , . They presented several bounds, and properties for the isolate domination number, and proposed the following problem(s).
Problems:(1) Characterize cubic graphs with .
(2) Characterize graphs with , or .
(3) Find bounds for .
In the following we state some known results that we need for the next. The corona graph of a graph , denoted by , is the graph obtained from by adding a pendant edge to every vertex of .
In this section we show that the decision problem for the isolate domination is NP-complete, even when restricted to bipartite graphs. We use a transformation from the 3-SAT problem. A truth assignment for a set of Boolean variables is a mapping . A variable is said to be true (or false) under if (or ). If is a variable in , then and are literals over . The literal is true under if and only if the variable is true under , and the literal is true if and only if the variable is false. A clause over is a set of literals over , and it is satisfied by a truth assignment if and only if at least one of its members is true under that assignment. A collection of clauses over is satisfiable if and only if there exists some truth assignment for that simultaneously satisfies all the clauses in . Such a truth assignment is called a satisfying truth assignment for . The 3-SAT problem is specified as follows.
Instance: A collection of clauses over a finite set of variables such that for .
Question: Is there a truth assignment for that satisfies all the clauses in ?