Star covers and star partitions of double-split graphs

Abstract

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.