In this article, we give two extended space formulations, respectively, for the induced tree and path polytopes of chordal graphs with vertex and edge variables. These formulations are obtained by proving that the induced tree and path extended incidence vectors of chordal graphs form Hilbert basis. This also shows that both polytopes have the integer decomposition property in chordal graphs. 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，分别提出了包含顶点变量与边变量的两种扩展空间表述。这些表述通过证明弦图的诱导树与路径扩展关联向量构成希尔伯特基而获得，同时表明在弦图中这两种多面体均具有整数分解性质。其中诱导树多面体的表述显然具有紧凑规模，而我们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。我们证明了哪些不等式定义出多面体的维面，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维面的不等式。对于某些图类，该超集包含冗余不等式。作为推论，我们得到在弦图类中，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。