The complete bipartite graph \(K_{1,3}\) 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 \(S\subseteq V(G)\) such that \(G-S\) 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.

完整的二部图\（K_ {1,3} \）被称为claw。有图有说是无爪，如果它不包含导出子同形爪。给定一个图G，NP硬图剥除问题（GDP）包括找到一个最小集\（S \ subseteq V（G）\）使得\（GS \）没有爪子。这项工作发展了GDP多面体的多面体研究，阐明了它的完整维度，提出并测试了五个方面的面：琐碎的不等式，爪子的不等式，恒星的不等式，灯笼的不等式和双星的不等式。总的来说，已经开发了四种具有分离启发法的分支剪切算法，以测试每个提出的族在随机图实例和随机间隔图实例上的计算优势。我们的结果表明，使用针对星型不等式提出的分离试探法的模型在几乎所有情况下都可以在两组实例上获得更好的结果。