SOLVING COLORING, MINIMUM CLIQUE COVER AND KERNEL PROBLEMS ON ARC INTERSECTION GRAPHS OF DIRECTED PATHS ON A TREE OLIVIER DURAND DE GEVIGNEY, FRED´ ERIC´ MEUNIER∗, CHRISTIAN POPA, JULIEN REYGNER, AND AYRIN ROMERO Abstract. Let T = (V; A) be a directed tree. Given a collection P of dipaths on T , we can look at the arc-intersection graph I(P;T ) whose vertex set is P and where two vertices are connected by an edge if the corresponding dipaths share a common arc. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs) and proved that they are perfect. DE graphs find one of their applications in the context of optical networks. For instance, assigning wavelengths to set of dipaths in a directed tree network consists in finding a proper coloring of the arc-intersection graph. In the present paper, we give • a simple algorithm finding a minimum proper coloring of the paths. • a faster algorithm than previously known ones finding a minimum multicut on a directed tree. It runs in O(jV jjPj) (it corresponds to the minimum clique cover of I(P;T )). • a polynomial algorithm computing a kernel in any DE graph whose edges are oriented in a clique-acyclic way. Even if we know by a theorem of Boros and Gurvich that such a kernel exists for any perfect graph, it is in general not known whether there is a polynomial algorithm (polynomial algorithms computing kernels are known only for few classes of perfect graphs). 1. Introduction Consider a directed tree T = (V; A) (with n vertices) and a collection of dipaths in this tree. The intersection graph I( ;T ) is a graph whose vertex set is and whereP two vertices are connected by an edge if the correspondingP dipaths share a common arc.P T with the collection is called a representation of I( ;T ). P These graphs generalize intervalP graphs, and also, as we will see, line-graphs of bipartite graphs. Monma and Wei, who started their study in a seminal paper on intersection graphs of paths on a tree, called them DE graphs (for directed edge path graphs) [16]. The purpose of their paper is an extensive study of the intersection graph by considering various situations: the tree could be directed or undirected, and the paths could be identified with their vertex set or their edge set. They proved various properties and explained how such graphs can be efficiently recognized. In particular, they explained how a tree T with a collection representing a given DE graph can be computed in polynomial time. P Afterwards, DE graphs received attention in papers with applications to optical fibers (see for instance [2, 3, 5, 8]), or from a combinatorial optimization point of view [6, 7]. The purpose of these studies consists often in the design of polynomial and efficient algorithms to compute various things, such as a minimum proper coloring of I( ;T ) (which can be interpreted as assignement of wavelengths with a minimum number of distinctP wavelengths) or a minimum clique cover of I( ;T ) (which is the minimum multicut problem). P The purpose of the present paper is to continue this quest of algorithmic properties of DE graphs. When working on a DE graph, we will always assume that a tree with a collection of dipaths ∗Corresponding author. 1 representing it is given. It is not a restriction in terms of polynomiality, since such a representation can be computed in polynomial time. Anyway, the time complexities will be expressed in terms of the size of the representation. 1.1. Definitions. Let G = (V; E) be a (undirected) graph. A stable set of G is a subset S V such that no edge of G connects two vertices of S. The stability number is the maximum size⊆ of a stable set of G and is denoted by α(G). A clique of G is a set of vertices any two of which are adjacent. The clique number is the maximum size of a clique and is denoted by !(G). A coloring of G is a color assignment on the vertices of G. If any two vertices connected by an edge get different colors (or equivalently, a partition of V into stable sets), we say that we have a proper coloring. The minimum number of colors in a proper coloring of G is called the chromatic number and is denoted by χ(G). A clique cover of G is a partition of V into cliques. The minimum number of cliques in a clique cover of G is called the clique cover number of G and is denoted byχ ¯(G). The following relations are immediate (recall that G¯ denotes the complementary graph of G, in which edges become non-edges and conversely): α(G) = !(G¯); χ(G) =χ ¯(G¯); α(G) χ¯(G);!(G) χ(G): ≤ ≤ G is said to be perfect if χ(H) = !(H) for all subgraphs H of G, or, equivalently, if α(H) =χ ¯(H) for all induced subgraphs H of G, since according to the well-known theorem of Lov´asz[12], G is perfect if and only if G¯ is perfect. Let D = (V; A) be a directed graph. Given a list of p ordered pairs of terminal vertices (si; ti), with i = 1; : : : ; p, a multicut is a subset of arcs intersecting, for each i = 1; : : : ; p, all dipaths going from si to ti. If D is a directed tree, note that for each ordered pair (si; ti), there is at most one dipath from si to ti. A cycle of length s in an oriented graph is a sequence of arcs (v1; v2); (v2; v3);:::; (vs; v1). A clique-acyclic orientation of G is an orientation G~ of the edges of G in such a way that each clique becomes acyclic. Said differently, there is no cycle of length 3 in the graph G~ . A kernel in an oriented graph D = (V; A) is a stable subset K V such that for each v V K, there is a w K such that (v; w) A. ⊆ 2 n 2 2 Let G = (V; E) be a bipartite graph with a total order v on the set of edges incident to v for each vertex v V .A matching M (a subset M E of disjoint edges) is said to be stable if 2 ⊆ whenever there is an edge e = uv E M, there is an edge f δ(u) M such that f u e or 2 n 2 \ f δ(v) M such that f v e, where δ(u) refers to the set of edges incident to the vertex u. We have2 the\ celebrated marriage theorem by Gale and Shapley. Theorem 1 (Stable marriages theorem, [9]). Let G = (V; E) be a bipartite graph with a total order v on the set of edges incident to v for each vertex v V . Then there is a stable matching in G and it can be computed in O( V E ). 2 j jj j 1.2. The intersection graph I( ;T ) as a perfect graph. Monma and Wei proved various results for DE graphs. To ease theP discussion, we summarize some of these results in a theorem. Theorem 2 (Monma and Wei [16]). DE graphs are perfect graphs, they can be recognized in poly- nomial time and, given a DE graph G, it is possible to compute in polynomial time a representation of it, i.e. a tree T and a collection of dipaths such that G = I( ;T ). P P Let = P1;:::;Pp . The source of Pi is denoted si and its target ti. In our context, α(I( ;T )) is theP maximumf numberg of arc-disjoint dipaths, !(I( ;T )) is the maximum number of dipathsP passing through a common arc, χ(I( ;T )) is the minimumP number of colors that can be assigned to the dipaths in such a way that two dipathsP sharing a common arc get different colors, andχ ¯(I( ;T )) 2 P P5 s1 = s4 s2 s3 = s6 P2 s7 = s8 P1 P3 s5 Pi := path si ti s9 → t1 s10 P6 P10 P4 t = t t3 = t4 t = t 2 5 8 10 P P9 8 P7 t6 = t7 = t9 I( ,T ) P χ(I( ,T )) = ω(I( ,T )) = 5 P P T α(I( ,T )) =χ ¯(I( ,T )) = 4 P P : a maximum stable set : a minimum multicut (= a minimum clique cover) Figure 1. Illustration of the perfectness of the (arc)-intersection graph of dipaths on a directed tree. is the minimum number of arcs intersecting all dipaths of , that is a minimum multicut of T with P the (si; ti) as ordered pairs of terminals. For the interpretation of !(I( ;T )) andχ ¯(I( ;T )), we use the fact that has the Helly property, noted first by Monma and Wei.P In the presentP context, P the Helly property means that, given a collection Pi1 ;:::;Pis of paths in such that they pairwise contain a common arc, then the whole collection contains a common arc.P The equality α =χ ¯ says that the maximum number of arc-disjoint dipaths equals the minimum multicut, and the equality ! = χ says that the maximum number of dipaths passing through a common arc is equal to the minimum colors that can be assigned to the dipaths in such a way that two dipaths sharing a common arc get different colors. Figure 1 is an illustration: the network on the left hand side is a representation of the DE graph on the right hand side; there are ten paths Pi, each of them is the unique dipath going from si to ti.