The Cycle-Path Indicator Polynomial of a Digraph

Abstract

A cycle-path cover of a digraph D is a spanning subgraph made of disjoint cycles and paths. In order to count such covers by types we introduce the cycle-path indicator polynomial of D. We show that this polynomial can be obtained by a deletion–contraction recurrence relation. Then we study some specializations of the cycle-path indicator polynomial, such as the geometric cover polynomial (the geometric version of the cover polynomial introduced by Chung and Graham), the cycle cover polynomial, and the path cover polynomial.As an application, we consider chromatic arrangements of non-attacking rooks and the associated polynomial which is symmetric and generalizes the usual rook polynomial.