Covering digraphs by paths

The problems of miniiiwn Lg.: ana minimum vertex covers by paths results relate to papers by Galki-M&ram, Meyniel, Alspach-Pullman and mait resulti is contimed with partially ordered sets. are discussed. The others. One of tie NM+oak The terminology is rather standard, generally following Berge [3]. In any case where ar&iguity may arise we give +finiticns. Graphs alnd digraphs are f@ite, except in Theorem 6, and have no loops or multiple edges. If multiple edges are allowed we use the term multigragh. Let G-(V, E) be a graph. k ,I ~&atr'o~r of G is a digralph whose underlyir-raph is G. If S is a set, ISI denotes its cardinality. If a set is sak to be maxir,,,m (minimum) it means that it is cardinality maximum (minimum). The order of G, that is, 1 V(G)I, is denoted by n. Also (E(G)1 := e. The ut!rtex ind~&r;lce number of G is the maximum cardinality of an independent set of vertioes. We ddnote it by &,(G). Let G be a digraph. DL(G) its dilkgraph is defined as follows: it is a digraph whose vertex set is E(G). There is an edge from a vertex x to a vertex :' if the terminal vertex of x and the initial vertdx of y coincide. We use the word path to mean a directed simple path. The length of a path is the number of edges it contains. r4 path of length two is called a couple. The two eel&a [a y], /[y, x] are also called the fwo-way edge joining x and y. The digraph obixti by deleting all two-way edges in DL(G), is denoted by DL(G). The edge inhpenderice number of G is defined by P,(G) = &(DL(G)). Given Z~L set A of vertices in G, denote by T(A) (resp. r(A)) the set of those irertices $1 hat are joined to (resp. fr3m) some vertex in A. Also T(A) = I " (A)U II-(A). If A is a singleton ,4 =1x}, we define d'(x) = II+({x})l, d-(x) = Ip'crx})i, a'(x)-d+(x)+ d-(x). Let A, B be two disjoint sets of vertices in G. We denote blr E(A) the set of edges in G both of whose endvertices belong to A. E(A, B) is the set of edges joining a vertex in A and a vertex in B. If B = V\A we call E(/i, B) the cut associated with A. By e(A), e(A, B) we denote …