k-L(2, 1)-labelling for planar graphs is NP-complete for k>=4

A mapping from the vertex set of a graph G = (V,E) into an interval of integers {0, . . . , k} is an L(2, 1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbour are mapped onto distinct integers. It is known that for any fixed k ≥ 4, deciding the existence of such a labelling is an NP-complete problem while it is polynomial for k ≤ 3. For even k ≥ 8, it remains NP-complete when restricted to planar graphs. In this paper, we show that it remains NP-complete for any k ≥ 4 by reduction from Planar Cubic TwoColourable Perfect Matching. Schaefer stated without proof that Planar Cubic Two-Colourable Perfect Matching is NP-complete. In this paper we give a proof of this. Key-words: L(2, 1)-labelling, distance contrained colouring, planar graph, complexity, channel assignment ∗ Brunel University, Kingston Lane, Uxbridge, UB8 3PH, UK. Supported by the EC Marie Curie programme NET-ACE (MEST-CT-2004-6724)[email protected] † projet Mascotte, I3S(CNRS and University of Nice-Sophia Antipolis) and INRIA, 2004 Route des Lucioles, BP 93, 06902 Sophia-Antipolis Cedex, France. Partially supported by the european project FET Aeolus. [email protected]. ‡ Brunel University, Kingston Lane, Uxbridge, UB8 3PH, UK. Partially supported by the Heilbronn Institute for Mathematical Research, Bristol,U.K. k-L(2, 1)-Labelling pour les graphes planaires est NP-Complete pour k ≥ 4. Résumé : Une application de l’ensemble des sommets d’un graphe G = (V,E) dans un intervalle des entiers naturels {0, . . . , k} est un L(2, 1)-labelling de G d’écart k si deux sommets adjacents recoivent des entiers à distance au moins 2 et deux sommets ayant un voisin en commun recoivent des entiers distincts. On sait que pour k ≥ 4, décider l’existence d’un L(2, 1)-labelling est un problème NP-complet pour k ≥ 4 alors que c’est polynomial pour k ≤ 3. Pour k ≥ 8 et pair, cela reste NP-complet restreint à la classe des graphes planaires. Dans ce rapport, nous montrons que cela reste NP-complet pour tout k ≥ 4 par réduction de Planar Cubic Two-Colourable Perfect Matching. Schaefer a affirmé sans preuve que ce problème est NP-complet. Nous en donnons une preuve ici. Mots-clés : L(2, 1)-labelling, coloration avec contrainte de distance, graphe planaire, complexité, allocation de fréquences k-L(2, 1)-Labelling for Planar Graphs is NP-Complete. 3 The Frequency Assignment Problem requires the assignment of frequencies to radio transmitters in a broadcasting network with the aim of avoiding undesired interference and minimising bandwidth. One of the longstanding graph theoretical models of this problem is the notion of distance constrained labelling of graphs. An L(2, 1)-labelling of a graph G is a mapping from the vertex set of G into the nonnegative integers such that the labels assigned to adjacent vertices differ by at least 2, and labels assigned to vertices at distance 2 are different. The span of such a labelling is the maximum label used. In this model, the vertices of G represent the transmitters and the edges of G express which pairs of transmitters are too close to each other so that an undesired interference may occur, even if the frequencies assigned to them differ by 1. This model was introduced by Roberts [11] and since then the concept has been intensively studied (see the survey of Yeh [13]). In their seminal paper, Griggs and Yeh [7] proved that determining the minimum span of a graph G, denoted λ2,1(G), is an NP-hard problem. Fiala et al. [5] proved that deciding λ2,1(G) ≤ k is NP-complete for every fixed k ≥ 4 and later Havet and Thomassé [8] proved that for any k ≥ 4, it remains NP-complete when restricted to bipartite graphs (and even a restricted family of bipartite graphs, i.e incidence graphs or first division of graphs). When the span k is part of the input, the problem is nontrivial for trees but a polynomial time algorithm based on bipartite matching was presented in [3]. The problem is still solvable in polynomial time if the input graph is outerplanar [9, 10]. Moreover, somewhat surprisingly, the problem becomes NP-complete for series-parallel graphs [4], and thus the L(2, 1)-labelling problem belongs to a handful of problems known to separate graphs of tree-width 1 and 2 by P/NP-completeness dichotomy. In this paper we consider the following problem. Problem 0.1 (Planar k-L(2, 1)-Labelling). Let k ≥ 4 be fixed. Instance: A planar graph G. Question: Is there an L(2, 1)-labelling with span k? Bodlaender et al. [1] showed that this problem is NP-complete if we require k ≥ 8 and k even. In the survey paper [2], it is suggested that the problem is NP-complete for all k ≥ 8 due to [6]. However this does not seem to be the case. In [6] there is a proof showing that the corresponding problem where k is specified as part of the input is NP-complete. This proof shows that the problem is NP-complete for certain fixed values of k. However it is far from clear for which values of k this is true. In this paper we first prove that Planar Cubic Two-Colourable Perfect Matching, which we define in the next section, is NP-Complete. This result was first stated by Schaefer [12] but without proof. In the second part of this paper we use this result in order to show that Problem 0.1 is NP-complete. 1 Preliminary results The starting problem for our reductions is Not-All-Equal 3SAT, which is defined as follows [12]. Definition 1.1 (Not-All-Equal 3SAT). Instance: A set of clauses each having three literals. Question: Can the literals be assigned value true or false so that each clause has at least one true and at least one false literal? In [12], it is shown that this problem is NP-complete. Our reduction involves an intermediate problem concerning a special form of two-colouring. In this section we define the intermediate problem and show that it is NP-complete. When k = 4 or k = 5, the final stage of our reduction is similar to the reduction in [5]. However we cannot use induction for higher RR n° 6840 4 N..Eggemann, F. Havet, and S. Noble