In this article, we give an extended space formulation for the induced tree and path polytopes of chordal graphs with variables associated with the edge and vertex sets. Whereas the formulation for the induced tree polytope is easily seen to have a compact size, the system we provide for the induced path polytope has an exponential number of inequalities. We show which of these inequalities define facets and exhibit a superset of the facet-defining ones that can be enumerated in polynomial time. We show that for some graphs, the latter superset contains redundant inequalities. As corollaries, we obtain that the problems of finding an induced tree or path maximizing a linear function over the edges and vertices are solvable in polynomial time for the class of chordal graphs .

翻译：本文针对弦图的诱导树与路径多面体，提出了基于边集与顶点集变量的扩展空间形式。其中诱导树多面体的形式显然具有紧凑规模，而本文为诱导路径多面体提供的系统则包含指数级数量的不等式。我们证明了其中哪些不等式定义面，并展示了一个可在多项式时间内枚举的超集，该超集包含所有定义面的不等式。研究表明对于某些图，该超集包含冗余不等式。作为推论，我们得出在弦图类中，寻找最大化边与顶点线性函数的诱导树或路径问题可在多项式时间内求解。