For a graph G with vertex set V, a set \(S\subseteq V\) is called a dominating set of G if every vertex in \(V-S\) has a neighbor in S. If the set \(S\subseteq V\) is a dominating set of G and the induced subgraph G[S] contains at least one isolated vertex, the set S is called an isolate dominating set of G. The minimum cardinality of a dominating set of G is called the domination number of G, denoted by \(\gamma (G)\), and the minimum cardinality of an isolate dominating set of G is called the isolate domination number of G, denoted by \(\gamma _0(G)\). It has been proved that the isolate domination problem is NP-complete in general. In this paper, we present a quadratic-time algorithm to compute the isolate domination number of a tree.

对于图ģ与顶点集V，一组\（S \ subseteq V \）被称为控制集的ģ如果在每个顶点\（VS \）具有在邻居小号。如果集合\（S \ subseteq V \）是一个控制集ģ和导出子ģ [小号]包含至少一种分离的顶点，该组小号称为分离支配集的ģ。控制集的最小基数ģ被称为控制数的ģ，记\（\伽马（G）\） ，和一分离控制集的最小基数ģ被称为分离控制数的ģ，记\（\伽马_0（G）\） 。已经证明孤立支配问题一般是NP完全的。在本文中，我们提出了一种二次时间算法来计算树的孤立支配数。