Characterizations of Postman Sets

Using results by McKee and Woodall on binary matroids, we prove that the set of postman sets has odd cardinality, generalizing a result by Toida on the cardinality of cycles in Eulerian graphs. We study the relationship between T-joins and blocks of the underlying graph, obtaining a decomposition of postman sets in terms of blocks. We conclude by giving several characterizations of T-joinswhicharepostmansetsand commenting on practical issues. Keywords: T-joins, postman sets, cardinality, graph block, decomposition 1 Basic Notations and Definitions We will consider undirected graphs G =(V,E). The set of odd degree vertices of G will be denoted by O(G), or simply by O when it is clear what the underlying graph is. Most other notations and conventions for graphs are similar to those in West [5]. In particular, paths and cycles have no repeated vertices, and loops are cycles. Given a subset T of vertices with |T | even, a set of edges J ⊂ E is a T-join if O(GJ) =T where GJ =(V,J). We will be interested in the family T of minimal T-joins: an inclusion-wise minimal T-join is just a T-join such that GJ is acyclic. Of course, T is a clutter, i.e. a family of subsets of some finite base set—here E—none of which is included in another. When T = ∅, the empty set is the unique minimal ∅-join, and it is convenient to work instead with the clutter of cycles (regarded as edge-sets) C, sothat every non-empty ∅-join may be written as a union of disjoint cycles. When T = O(G), the minimal T-joins are called postman sets, and we will indicate the corresponding clutter by P. We observe that although there are always postman sets, perhaps only the empty set (i.e. P = {∅}), we may have T = ∅ if some connected component of G contains an odd number of vertices of T. Similarly, C could be empty.

Characterizations of postman sets

Using results by McKee and Woodall on binary matroids, we show that the set of postman sets has odd cardinality, generalizing a result by Toida on the cardinality of cycles in Eulerian graphs. We study the relationship between T-joins and blocks of the underlying graph, obtaining a decomposition of postman sets in terms of blocks. We conclude by giving several characterizations of T-joins which are postman sets.Fil: Aguilera, Néstor Edgardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Leoni, Valeria Alejandra. Universidad Nacional de Rosario; Argentin