Operations Research Letters Operations Research Letters 32 (2004) 547–556 www.elsevier.com/locate/dsw Enumerating maximal independent sets with applications to graph colouring Jesper Makholm Byskov 1 BRICS 2, Department of Computer Science, University of Aarhus, IT-parken, Aabogade 34, DK-8200 Aarhus N, Denmark Received 4 February 2004; accepted 3 March 2004 Abstract We give tight upper bounds on the number of maximal independent sets of size k (and at least k and at most k) in graphs with n vertices. As an application of the proof, we construct improved algorithms for graph colouring and computing the chromatic number of a graph. c 2004 Elsevier B.V. All rights reserved. Keywords: Maximal independent set; Graph colouring; Chromatic number; Graph algorithms; Extremal graphs 1. Introduction 1.1. Maximal independent sets Algorithms for ÿnding maximal independent sets Finding MISs in a graph is a well-studied prob- (MISs) and MISs of size k (k-MISs) in graphs can lem arising naturally in many ÿelds. For a survey, see be used in algorithms for colouring graphs. We show Bomze et al. [1, Chapter 5.1]. upper bounds on the number of MISs and k-MISs in The maximum number of MISs in a graph with undirected graphs with n vertices and give algorithms n vertices is the same as the maximum number of for ÿnding them all. We also show how to use these maximal cliques in a graph with n vertices by look- algorithms to develop faster algorithms for deciding ing at the complementary graph with edges between k-colourability of a graph and ÿnding the chromatic all non-neighbours in the original graph. Miller and number of a graph. Muller [11] were the ÿrst to show the so-called Moon and Moser bound, which was independently dis- covered by Moon and Moser [12], namely that a graph can have at most 3n=3 MISs and that there are This paper is an extended version of a short paper appearing in Proceedings of the 14th Symposium on Discrete Algorithms graphs having this many, the so-called Moon and (SODA) 2003 [3]. It also contains an algorithm for ÿnding the Moser graphs (see Theorem 1). The proofs in both chromatic number of a graph from a technical report [2]. papers are combinatorial, but there exists quite a few 1 Part of this paper was written while visiting University of algorithms for ÿnding all MISs in a graph, for exam- California, Irvine. ple the algorithm by Tsukiyama et al. [14] which runs 2Basic Research in Computer Science (www.brics.dk), funded by the Danish National Research Foundation. in time within a polynomial factor of the number of E-mail address: [email protected] (J.M. Byskov). MISs in the graph. 0167-6377/$ - see front matter c 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2004.03.002 548 J.M. Byskov / Operations Research Letters 32 (2004) 547–556 For MISs of size at most k, Eppstein [7] has removed vertex. Using that idea to branch on vertices shown that no graph contains more than 34k−n4n−3k with degree at least four and a complicated case analy- of these and he has also given an algorithm for sis for the remaining graphs, Schiermeyer gets his run- ÿnding them all in time within a polynomial fac- ning time. He also gives algorithms for 4-, 5- and tor of this bound. For k = n=3, Eppstein’s bound is 6-colouring, but they do not have the running times the same as the bound of Moon and Moser, but for that he claims [10]. k 6 n=3 it is smaller. Croitoru [5] showed that in The current best algorithm for 3-colouring is by a graph with no MIS larger than k, the number of Eppstein [6] and runs in time O(1:3289n). The al- MISs is at most n=k(n=k+1)k−n(n=k +1)n−n=kk . gorithm ÿrst transforms the graph and then solves For n=4 6 k 6 n=3 this expression is the same as the a generalised version of 3-colouring for the trans- bound of Eppstein. formed graph. Both parts require a rather involved In this paper, we extend the proof of Moon and case analysis. In the same paper, Eppstein gives ran- Moser to show that the bound of Croitoru is an upper domised algorithms for solving generalisations of 4- bound on the number of k-MISs for all k. Similar to and 5-colouring in time O(1:8072n) and O(2:2590n), Moon and Moser, we show that the bound is tight and respectively. These are the previous best algorithms that the extremal graphs are unique. For MISs of size for those problems. at most k the bound of Croitoru holds for k 6 n=3 and The algorithm by Lawler for ÿnding the chromatic for k ¿ n=3 the bound of Moon and Moser is a tight number has also been improved by Eppstein [7] who upper bound. For MISs of size at least k the bound gave an algorithm running in time O(2:4151n) still us- of Croitoru holds for k ¿ n=3 and the bound of Moon ing space O(log n·2n). His algorithm is also described and Moser is tight for k 6 n=3. We also extend the in Section 4. algorithm of Eppstein to ÿnd all k-MISs, MISs of size In this paper, we give new algorithms for 4-, 5- and at most k or MISs of size at least k in time within 6-colouring using polynomial space and running in a polynomial factor of our bounds on the number of time O(1:7504n), O(2:1592n) and O(2:3289n), respec- these. tively. We also give algorithms for ÿnding all maxi- Finally, we give an algorithmic proof that the num- mal 3-colourable subgraphs and for 6-colouring both ber of MISs in a triangle-free graph is at most 2n=2,a running in time O(2:2680n), but using space O(2n). result proven earlier by Hujter and Tuza [8] using a For ÿnding all maximal k-colourable subgraphs for combinatorial argument. k ¿ 4, we give algorithms running in time O(2:4023n) and space O(2n). We use these algorithms in a new 1.2. Colouring algorithm for ÿnding the chromatic number in time O(2:4023n) using space O(2n). The ÿrst algorithms with non-trivial running times for colouring are by Lawler [9] who gave an O(1:4423n)-time algorithm for 3-colouring, an 2. Preliminaries O∗(2n)-time algorithm for 4-colouring (for a deÿni- tion of O∗, see Section 2) and an O(2:4423n)-time All graphs in this paper are simple, undirected algorithm for ÿnding the chromatic number using graphs G =(V; E) and we let n = |V |. The subgraph space O(log n · 2n). The last algorithm builds on pre- induced by a subset of the vertices S ⊆ V is denoted vious work of Christoÿdes [4]. All three algorithms G[S] and we let N(v)={w ∈ V | (v; w) ∈ E} denote are described in Section 4. the set of neighbours of v and NL (v)={v}∪N(v). Schiermeyer [13] gave an algorithm for 3-colouring The graph Kn is the complete graph on n vertices. running in time O(1:398n). The main idea is that a pair We let I(G) denote the set of all MISs in G; I 6k (G) 6k of vertices must either have diKerent colours in which the subset of those that have size at most k and Iv (G) case an edge between them can be added or have the the subset of those that contain v. Similarly, we deÿne =k =k ¿k ¿k same colour in which case they can be contracted, I (G), Iv (G), I (G) and Iv (G). meaning that one of them is removed and edges are All running times in this paper are exponential and added between the other and all the neighbours of the we ignore polynomial factors. We use the notation J.M. Byskov / Operations Research Letters 32 (2004) 547–556 549 O∗(cn) to denote the class of functions that are within a polynomial factor of cn. If we round up c we can omit the star, as a polynomial times an exponential function is slow growing than any exponential function with a larger exponent. We only state the space usage, which we count in bits, when it is not polynomial. 3. Maximal independent sets Fig. 1. Construction of Gv→w. Miller and Muller [11] and Moon and Moser [12] independently gave the following upper bound on the graphs. The k-MISs in Gv→w containing v are exactly number of MISs in a graph: the same as those containing w, with v replacing w,so =k =k =k =k Theorem 1(Miller and Muller [11], Moon and Moser |I (Gv→w)| = |I (G)| + |Iw (G)|−|Iv (G)| [12]). The maximum number of MISs in any graph is + ···: n=3 3 if n ≡ 0 (mod 3); All k-MISs in G that neither contain v nor w are 4 · 3(n−4)=3 if n ≡ 1 (mod 3); (1) also k-MISs in G : they are independent since all v→w 2 · 3(n−2)=3 if n ≡ 2 (mod 3); changes involve only edges containing v and they are maximal since v cannot be added in G as it is a and the extremal graphs consist of a union of either v→w copy of w.