Given a weighted undirected graph~$G$ and a nonnegative integer~$k$, the maximum~$k$-star colorable subgraph problem consists of finding an induced subgraph of~$G$ which has maximum weight and can be star colored with at most~$k$ colors; a star coloring does not color adjacent nodes with the same color and avoids coloring any 4-path with exactly two colors. In this article, we investigate the polyhedral properties of this problem. In particular, we characterize cases in which the inequalities that appear in a natural integer programming formulation define facets. Moreover, we identify graph classes for which these base inequalities give a complete linear description. We then study path graphs in more detail and provide a complete linear description for an alternative polytope for~$k=2$. Finally, we derive complete balanced bipartite subgraph inequalities and present some computational results.
Citation
TU Darmstadt, Department of Mathematics, Research Group Optimization, Dolivostr. 15, 64293 Darmstadt, January 2015