The Induced Star Partition of Graphs

Abstract

Given a graph G, we call a partition \((V_1, V_2, \ldots , V_k)\) of its vertex set an induced star partition of G if each induced subgraph \(G[V_i]\) is isomorphic to a \(K_{1,r}\), \(r\ge 0\). In this paper, we consider the problem of partitioning a graph into a minimum number of induced stars and its decision versions. This problem may be viewed as an amalgamation of the well-known dominating set problem and coloring problem. Although this problem coincides with the dominating set problem on \(K_3\)-free graphs, it resembles, in its hardness, the coloring problem on general graphs. We establish the following results: (1) Deciding whether a graph can be partitioned into k induced stars is NP-complete for each fixed \(k\ge 3\) and has a polynomial time algorithm for each \(k\le 2\). (2) It is NP-hard to approximate the minimum induced star partition size within \(n^{1-\epsilon }\) for all \(\epsilon > 0\). (3) The decision version of the induced star partition problem is NP-complete for (a) subcubic bipartite planar graphs, (b) line graphs (a subclass of \(K_{1,r}\)-free graphs (\(r\ge 3\))), (c) \(K_{1,5}\)-free split graphs and (d) co-tripartite graphs. (4) The minimum induced star partition problem has (a) an \(\frac{r}{2}\)-approximation algorithm for \(K_{1,r}\)-free graphs (\(r\ge 2\)) and (b) a 2-approximation algorithms for split graphs.