Declawing a graph: polyhedra and Branch-and-Cut algorithms

The complete bipartite graph K1,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{1,3}$$\end{document} is called a claw. A graph is said to be claw-free if it contains no induced subgraph isomorphic to a claw. Given a graph G, the NP-hard Graph Declawing Problem (GDP) consists of finding a minimum set S⊆V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S\subseteq V(G)$$\end{document} such that G-S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G-S$$\end{document} is claw-free. This work develops a polyhedral study of the GDP polytope, expliciting its full dimensionality, proposing and testing five families of facets: trivial inequalities, claw inequalities, star inequalities, lantern inequalities, and binary star inequalities. In total, four Branch-and-Cut algorithms with separation heuristics have been developed to test the computational benefits of each proposed family on random graph instances and random interval graph instances. Our results show that the model that uses a separation heuristics proposed for star inequalities achieves better results on both set of instances in almost all cases.