A graph that is isomorphic to the complete bipartite graph \(K_{1,r}\) for some \(r\ge 0\) is called a star. A collection \(\mathcal {C} = \{V_1, \ldots , V_k\}\) of subsets of the vertex set of a graph \(G = (V, E)\) is called a star cover of G if each set in the collection induces a star and has \(V_1\cup \ldots \cup V_k = V\). A star cover \(\mathcal {C}\) of a graph \(G = (V, E)\) is called a star partition of G if \(\mathcal {C}\) is also a partition of V. The problem Star Cover takes a graph G as input and asks for a star cover of G of minimum size. The problem Star Partition takes a graph G as input and asks for a star partition of G of minimum size. From Shalu et al. (Discrete Appl Math 319:81–91, 2022), it follows that both these problems are NP-hard even for bipartite graphs. In this paper, we show that both Star Cover and Star Partition have \(O(n^7)\) time exact algorithms for double-split graphs. Proving that our algorithms indeed have running time \(\varOmega (n^7)\) necessitates the construction of an intricate infinite family of double-split graphs meeting several requirements. Other contributions of the paper are a simple linear time recognition algorithm for double-split graphs and a useful succinct matrix representation for double-split graphs.

对于某些\(r\ge 0\)来说，与完全二部图\(K_{1,r}\)同构的图称为星形。图\(G = (V, E)\)的顶点集子集\(\mathcal {C} = \{V_1, \ldots , V_k\}\)称为G的星型覆盖，如果集合中的每个集合都会产生一颗星并具有\(V_1\cup \ldots \cup V_k = V\)。如果 \(\mathcal {C}\)也是 V 的划分，则图\(G = (V, E)\)的星形覆盖\(\mathcal {C}\)称为G的星形划分。星形覆盖问题以图G作为输入，并要求G的最小尺寸的星形覆盖。星形分区问题以图G作为输入，并要求G的最小尺寸的星形分区。来自沙鲁等人。 (Discrete Appl Math 319:81–91, 2022)，因此即使对于二部图，这两个问题也是 NP 困难的。在本文中，我们展示了星型覆盖和星型分区对于双分割图都有\(O(n^7)\)时间精确算法。证明我们的算法确实具有运行时间\(\varOmega (n^7)\)需要构建满足多个要求的复杂的无限双分裂图族。该论文的其他贡献是用于双分裂图的简单线性时间识别算法和用于双分裂图的有用的简洁矩阵表示。