Star Covers and Star Partitions of Cographs and Butterfly-free Graphs

Abstract

A graph that is isomorphic to $K_{1,r}$ for some $r\ge 0$ is called a star. For a graph $G = (V, E)$, any subset S of its vertex set V is called a star of G if the subgraph induced by S is a star. A collection $\mathcal {C} = \{V_1, \ldots , V_k\}$ of stars in G is called a star cover of G if $V_1\cup \ldots \cup V_k = V$. A star cover $\mathcal {C}$ of G is called a star partition of G if it is also a partition of V. Given a graph G, the problem Star Cover asks for a star cover of G of minimum size. Given a graph G, the problem Star Partition asks for a star partition of G of minimum size. Both the problems are NP-hard even for bipartite graphs [24]. In this paper, we obtain exact $O(n^2)$ time algorithms for both Star Cover and Star Partition on $(C_4,P_4)$-free graphs and on $(2K_2,P_4)$-free graphs. We also prove that Star Cover and Star Partition are polynomially equivalent, up to the optimum value, for butterfly-free graphs and present an $O(n^{14})$ time $O(\log n)$-approximation algorithm for these equivalent problems on butterfly-free graphs. We also obtain $O(\log n)$-approximation algorithms for Star Cover on hereditary graph classes.

The second author is supported by DST-SERB MATRICS: MTR/2022/000870.