Extended formulations for induced tree and path polytopes of chordal graphs

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.