Abstract. The main result of this paper is the introduction of marked graphs and the marked graph polynomials ([Formula: see text]-polynomial) associated with them. These polynomials can be defined via a deletion-contraction operation. These polynomials are a generalization of the [Formula: see text]-polynomial, introduced by Noble and Welsh, and a specialization of the [Formula: see text]-polynomial, introduced by Ellis-Monaghan and Moffatt. In addition, we describe an important specialization of the [Formula: see text]-polynomial, which we call the [Formula: see text]-polynomial. Furthermore, we present an efficient algorithm for computing the chromatic symmetric function of a graph in the star basis of symmetric functions. As an application of these tools, we prove that proper trees of diameter at most 5 are reconstructible from its chromatic symmetric function.

A flag matroid can be viewed as a chain of matroids linked by quotients. Flag matroids, of which relatively few interesting families have previously been known, are a particular class of Coxeter matroids. In this paper we give a family of flag matroids arising from an enumeration problem that is a generalization of the tennis ball problem. These flag matroids can also be defined in terms of lattice paths, and they provide a generalization of the lattice path matroids of [J. Bonin, A. de Mier, and M. Noy, J. Combin. Theory Ser. A, 104 (2003), pp. 63–94].