V-order is a global order on strings related to Unique Maximal Factorization Families (UMFFs), themselves generalizations of Lyndon words. V-order has recently been proposed as an alternative to lexicographic order in the computation of suffix arrays and in the suffix-sorting induced by the Burrows-Wheeler transform. Efficient V-ordering of strings thus becomes a matter of considerable interest. In this paper we discover several new combinatorial properties of V-order, then explore the computational consequences; in particular, a fast, simple on-line V-order comparison algorithm that requires no auxiliary data structures. (C) 2016 Elsevier B.V. All rights reserved.