Approximation algorithms for finding and partitioning unit-disk graphs into co-k-plexes B. Balasundaram∗, S. S. Chandramouli†and S. Trukhanov‡
Abstract This article studies a degree-bounded generalization of independent sets called co-k-plexes. Constant factor approximation algorithms are developed for the maximum co-k-plex problem on unit-disk graphs. The related problem of minimum co-k-plex coloring that generalizes classical vertex coloring is also studied in the context of unit-disk graphs. We extend several classical approximation results for independent sets in UDGs to co-k-plexes, and settle a recent conjecture on the approximability of co-k-plex coloring in UDGs. Key-words: unit-disk graph; independent set; graph coloring; co-k-plex; k-dependent set; defective coloring; t-improper coloring
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Introduction
A simple approach to modeling the connectivity and interference characteristics of a wireless network is using Unit-Disk Graphs (UDGs). In such graphs, the vertex set V represents the set of wireless nodes with associated disk centers cv for each v ∈ V located in the Euclidean plane. Under the proximity model for UDGs, an edge (u, v) ∈ E exists between vertices u and v if the Euclidean distance between the centers of corresponding disks is within a specified proximity threshold ρ, that is k cu − cv k≤ ρ. Mathematically this is equivalent under scaling to intersection graphs of unit circles in the Euclidean plane. Unlike many other geometric intersection graph models, UDGs are not necessarily perfect or planar as a cycle on five vertices and a complete graph on five vertices can both be represented as UDGs. Given the geometric representation (centers and proximity threshold), the UDG G = (V, E) is determined. However given a graph G, deriving a geometric representation or concluding none exists is NP-hard (5). For this reason, in this article we assume that the UDG is given with its geometric representation. Given a graph, a clique is a subset of pairwise adjacent vertices, an independent set is a subset of pairwise nonadjacent vertices, and a proper coloring partitions the vertex set into independent sets. These three ideas are extensively used in wireless network applications as an edge in the UDG model indicates the possibility of interference. Hence, an independent set corresponds to wireless nodes that can transmit simultaneously and is used to find broadcasting sets (26). A proper coloring is used in frequency assignment problems to identify the number of frequency bands required for interference free communication as each partition can be assigned the same frequency band (16). A clique, on the other hand, corresponds to a set of wireless nodes where transmission from any one node could be received by all others. For this reason, this approach is typically used to cluster wireless nodes by partitioning the UDG into cliques (24). ∗
School of Industrial Engineering & Management, Oklahoma State University, Stillwater, OK 74078, USA,
[email protected] † Indian Institute of Technology-Madras, Chennai 600036, India,
[email protected] ‡ Microsoft Corporation, One Microsoft Way, Redmond, WA 98052-6399.
[email protected]
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Cliques and independent sets are equivalent under graph complementation, and they typically share complexity and inapproximability results for arbitrary graphs (15; 18). However for UDGs, the maximum clique problem is polynomial time solvable while the maximum independent set problem is NP-hard (7), which is in curious analogy to planar graphs. The related minimum graph coloring problem (7), and the minimum clique partitioning problem (6) on UDGs are also known to be NP-hard. Hence, the research in this area has emphasized approximation algorithms and distributed heuristics. Recent surveys on these issues are available in (3; 13; 14). In this article, we are motivated by applications in communication networks that permit relaxations of these classical models to be used. For instance, if a receiver can distinguish interference from limited number of sources while extracting the desired signal, then it could be part of a set in which it has a limited number of neighbors instead of none. One such application in satellite communications is discussed in (19; 23; 20), where it would be sufficient to find a set of nodes that induce a subgraph of bounded maximum degree, where this bound is specified by the user or determined by the application. The remainder of this article is organized as follows. In Section 2, we provide formal definitions of all the relevant concepts and introduce the notations we use in this article. A review of relevant literature is provided and our key contributions are identified in Section 3. Approximation algorithms for the generalizations of independent sets and coloring in UDGs are presented in Sections 4 and 5. We close with a summary in Section 6.
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Definitions and Notations
We always consider simple, finite, undirected graphs denoted by G = (V, E). Recall that we denote the corresponding disk center of a vertex v ∈ V by cv ∈ R2 and the proximity threshold for the UDG model by ρ. For a vertex v ∈ V , N (v) is its neighborhood and N [v] = N (v) ∪ {v} is its closed ¯ = (V, E) ¯ and the subgraph induced by neighborhood. We denote the complementary graph by G S ⊆ V by G[S]. We denote by δ(G) and ∆(G) the minimum and maximum vertex degrees in G, respectively. A clique C is a subset of vertices such that for all i, j ∈ C, we have (i, j) ∈ E. The maximum clique problem is to find a largest cardinality clique in G and the clique number ω(G) is the size of a maximum clique. An independent set I is a subset of vertices such that for all i, j ∈ I, (i, j) ∈ / E. As before, the maximum independent set problem is to find an independent set of maximum cardinality. The independence number of a graph G is denoted by α(G) and it is the size of a maximum independent set. A maximal clique (independent set) is one that is not a proper subset of another clique (independent set). Note that C ⊆ V is a clique in G if and only if C is an ¯ and hence, ω(G) = α(G). ¯ independent set in G A proper coloring of a graph is one in which every vertex is colored such that no two vertices of the same color are adjacent. A graph is said to be t-colorable if it admits a proper coloring with t colors. Under proper coloring, the vertices of the same color referred to as a color class, induce an independent set. The chromatic number of the graph χ(G), is the minimum number of colors required to properly color G. Note that for any graph G, ω(G) ≤ χ(G), as different colors are required to color the vertices of a clique. A related problem is the minimum clique partitioning problem, which is to partition the given graph G into a minimum number of cliques, χ(G). ¯ Note ¯ and χ(G) ¯ that this is exactly the graph coloring problem on G ¯ = χ(G). In this article, for notational convenience, we denote all graph invariants of induced subgraphs in the following manner. The independence number of G[S] for instance is denoted by α(S) instead of α(G[S]). Next we describe the generalizations of interest, provide a background and a summary
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of relevant results from literature. Definition 1 ((27)). Given a graph G = (V, E), a subset S ⊆ V is a k-plex if |N (v) ∩ S| ≥ |S| − k ∀ v ∈ S, that is δ(S) ≥ |S| − k. Definition 2. Given a graph G = (V, E), a subset J ⊆ V is a co-k-plex if |N (v) ∩ J| ≤ k − 1 ∀ v ∈ J, that is ∆(S) ≤ k − 1. Note that 1-plexes and co-1-plexes are simply cliques and independent sets, respectively. For k > 1, the clique definition is relaxed by a k-plex as it allows for at most k − 1 non-neighbors in S, while the co-k-plex is a relaxation of independent set definition as it allows for at most k − 1 ¯ The maximum neighbors in J. Clearly, S is a k-plex in G if and only if S is a co-k-plex in G. co-k-plex problem is to find a largest cardinality co-k-plex in G, the cardinality of which is the co-k-plex number of G denoted by αk (G). The maximum k-plex problem is similarly defined with the k-plex number denoted by ωk (G). Maximal k-plexes and co-k-plexes are also defined similar to cliques and independent sets by inclusion. A natural partitioning extension of this relaxation is the following. Definition 3. A proper co-k-plex coloring of a graph G = (V, E) is an assignment of colors to V such that each vertex has at most k − 1 neighbors in the same color class. Clearly, co-1-plex coloring is classical graph coloring and this model offers a relaxation for k > 1. Note that co-k-plex coloring aims to partition the given graph into co-k-plexes and hence it is equivalent to k-plex partitioning on the complement graph. Recall the motivating application introduced in Section 1 for which the maximum co-k-plex problem and co-k-plex coloring using minimum number of colors are relevant. However, originally the k-plex model was introduced in (27) to model “cohesive subgroups” in social network analysis. The overly restrictive and impractical definition of cliques motivated the development of this and other clique relaxations in this field. The co-k-plex was introduced as the complementary structure of k-plex, and related co-k-plex coloring and k-plex partitioning were defined and studied in (2). For a detailed discussion on the advantages of k-plex as a cluster model see (4; 2). Interestingly, the concepts of co-k-plex and co-k-plex coloring were independently developed in literature prior to the work of (2), seemingly with no connection to k-plexes, and naturally under different nomenclature. A brief summary of the relevant results from literature follows.
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Previous Work and Our Contributions
The maximum co-k-plex problem was introduced under the name of “maximum k-dependent set problem” in (12) and the decision version was shown to be NP-complete on arbitrary graphs. A co-k-plex is precisely a (k − 1)-dependent set. The study was furthered in (11) where among other results, the problem was shown to be NP-complete even for planar and bipartite graphs for fixed k ≥ 2. Note that the maximum co-1-plex problem (independent set) problem is polynomial time solvable on bipartite graphs through graph matching techniques (8). Recently in (23; 20), the problem is shown to be NP-complete for every fixed k on UDGs and is shown to admit a polynomial 3
time approximation scheme (PTAS) on UDGs. However, no simple constant factor approximation algorithms are available for this problem on UDGs. In this article, we develop the first constant factor approximation algorithm for this problem that is much simpler, quicker and distributable, when compared to the PTAS which offers any desired approximation guarantee. It should be noted that the PTAS, similar to the well-known approach for independent sets in UDGs (22), is more complicated requiring exact solution of the problem by complete enumeration on smaller squares into which the UDG is subdivided on the Euclidean plane. The pursuit of a constant factor approximation algorithm also led us to extend the classical work of Marathe et al. (25) on approximating independent sets in UDGs and Hochbaum (21) on approximating independent sets in claw-free graphs. The minimum co-k-plex coloring problem has been studied under the name “defective coloring” (9; 17; 1; 10) and “improper coloring” (19; 23; 20) in the literature. The decision version of minimum co-k-plex coloring problem is also NP-complete for every fixed k on UDGs (23; 20). Furthermore, a polynomial time 6-approximation algorithm for the problem on UDGs is developed in (23; 20). Havet et al. (20) conjecture the existence of a polynomial time 3-approximate algorithm for this problem on UDGs. This conjecture is settled in this article by extending a classical bound on the chromatic number of a graph due to Szekeres and Wilf (28).
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Co-k-plexes in Unit-Disk Graphs
Proposition 1. For any simple graph G = (V, E) the co-k-plex number αk (G) and independence number α(G) are related as αk (G) ≤ kα(G). Proof. Suppose α(G) = m and αk (G) ≥ km + 1. We will find an independent set I in G of size at least m + 1, thereby deriving a contradiction. Let G(0) be a co-k-plex induced in G of size km + 1. Apply the following greedy algorithm for a maximal independent set in the graph G(0) , with set I initially empty. Pick any vertex v from V (G(i) ), let I ← I ∪ {v} and G(i+1) ← G(i) − N [v]. Increment i and repeat until G(i+1) is null. Since G(0) is a co-k-plex, so are all G(i) , and thus ∆(G(i) ) ≤ k − 1. So |V (G(i+1) )| = |V (G(i) )| − 1 − |N (v) ∩ V (G(i) | ≥ |V (G(i) )| − k. After m iterations |I| = m and |V (G(m) )| ≥ km + 1 − km = 1. So the algorithm can add at least one more vertex to independent set I, contradicting the initial assumption. Corollary 1. For any simple graph G = (V, E) the k-plex number ωk (G) and clique number ω(G) are related as ωk (G) ≤ kω(G). Remark 1. Note that the bound is tight as ∀k, m > 0 there exists a graph G with α(G) = m and αk (G) = km. One of the simplest example of such graphs is the disjoint union of m cliques of size k, which also has a UDG representation. Remark 2. Note that the 3-approximate algorithm for independent sets in UDGs (25) together with Proposition 1 implies a 3k-approximate algorithm for co-k-plexes in UDGs. The following results provide a more direct proof of existence of a 3k-approximate algorithm for co-k-plexes on UDGs by extending many interesting intermediate results from Marathe et al. (25) and Hochbaum (21). Corollary 2. Let G = (V, E) be a UDG. For any v ∈ V , the co-k-plex number αk (N (v)) ≤ 5k.
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{v2k+1 ,…,v3k } {vk+1 ,…,v2k }
{v3k+1 ,…,v4k } {v1 ,…,vk }
{v4k+1 ,…,v5k } Figure 1: Sharpness of Corollary 2. Proof. By Proposition 1 we have, αk (N (v)) ≤ kα(N (v)) ≤ 5k where the last inequality follows from the result of Marathe et al. (25) that α(N (v)) ≤ 5. Remark 3. Consider the proximity model of the UDG in Figure 1 with vertex set V = {v0 , v1 , . . . , v5k } and required proximity ρ. Then |N (v0 )| = |V \ {v0 }| = 5k with five groups of k vertices, with their centers inside five sectors of angle φ each and at distance ρ. Clearly, we can choose θ > π3 and φ > 0 satisfying 5(θ+φ) = 2π, so that the vertices in each φ-sector forms a co-k-plex and there is no edge between two φ-sectors. Thus we have αk (N (v0 )) = 5k indicating the sharpness of Corollary 2. Corollary 3. Let G = (V, E) be a UDG. Let v ∈ V be a vertex corresponding to minimum Xcoordinate, then the co-k-plex number αk (N (v)) ≤ 3k. Corollary 3 is also a sharp extension of the result in (25) and can be proved along similar lines as Corollary 2. Definition 4. A graph G = (V, E) is said to be (p + 1; k)-claw free if for every v ∈ V , αk (N (v)) ≤ p. A polynomial time p-approximation algorithm for finding independent sets in (p + 1; 1)-claw free graphs has been developed by Hochbaum (21), which has been applied to UDGs by Marathe et al. (25). Note that UDGs are are (6; 1)-claw free. The following results extend these ideas for co-k-plexes. Proposition 2. Let G = (V, E) be a (p + 1; k)-claw free graph and let J be a maximal co-k-plex in G. Then, αk (G) |J| ≥ . p 5
Proof. Suppose J ∗ is a maximum co-k-plex in G. Further assume that p ≥ k, as the only (p + 1; k)claw free graphs with p < k are themselves co-k-plexes. Then [ J∗ \ N [v] = ∅, v∈J
otherwise there exists a v ∈ J ∗ such that N [v] ∩ J = ∅ contradicting the maximality of J. Hence we have, X |J ∗ | = αk (G) ≤ |N [v] ∩ J ∗ | ≤ p|J|. v∈J
Corollary 4. There exists a polynomial time p-approximate algorithm for the maximum co-k-plex problem in (p + 1; k)-claw free graphs, implying a 5k-approximate algorithm for unit-disk graphs. The approximation ratio can be improved easily by noting Corollary 3 as stated in the following result. Theorem 1. There exists a polynomial time 3k-approximate algorithm for finding co-k-plexes in a unit-disk graph G = (V, E). Proof. Consider the following algorithm that returns a maximal co-k-plex. Initialize J to be empty and V ′ to V . Pick vertex v corresponding to the minimum X-coordinate (left most disk); if J ∪ {v} is a co-k-plex in G then add v to J; delete v from V ′ and repeat until V ′ is empty. Denote the vertices in J by v1 , . . . , vq in the order in which they were added to J. If J ∗ denotes a maximum co-k-plex as before, we have q [ |J ∗ | ≤ N [vi ] ∩ J ∗ . i=1
By Corollary 3, |N [v1 ] ∩ J ∗ | ≤ 3k. Note that each vi is the left most vertex in the residual graph G[V ′ ] in each iteration. Hence for 1 < i ≤ q, the number of neighbors of vi in J ∗ located to the right of vi is at most 3k. The nodes in J ∗ that are to the left of vi have already been counted in the sum over v1 , . . . , vi−1 . Hence each term in that summation can be replaced by 3k resulting in the claim αk (G) ≤ 3k|J|.
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Co-k-plex Coloring of Unit-Disk Graphs
Definition 5. Denote by d(G), the largest d such that G has an induced subgraph of minimum degree at least d. The invariant d(G) can be found in polynomial time using a simple algorithm (21) along with an associated ordering v1 , . . . , vn of vertices of G such that any vi has at most d(G) neighbors in v1 , . . . , vi−1 . It is a well known result in classical graph coloring that χ(G) ≤ d(G) + 1 due to Szekeres and Wilf (28). The following result extends this observation to co-k-plex coloring. Proposition 3. For any graph G, the following identity is true and there exists a polynomial time algorithm to find a co-k-plex coloring that uses at most ⌊ d(G) k ⌋ + 1 colors. χk (G) ≤ ⌊
d(G) ⌋ + 1. k
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Proof. Consider the following algorithm which returns a proper co-k-plex coloring of G. Process the vertices according to the ordering as described in Definition 5; in that order, for each vertex assign the smallest available color. A color is said to be available at a vertex if it has been assigned to at most k − 1 neighbors of the current vertex, already processed in the list ordering. Consider any vertex vi , i = 2, . . . , n. It can have at most d(G) neighbors already colored in the list so far. There must exist a color from the set {1, . . . , ⌊ d(G) k ⌋ + 1} that has been assigned to at most k − 1 neighbors of vi . If not, the number of colored neighbors of vi exceeds d(G). Since this is true for any vi , the above algorithm uses no more than ⌊ d(G) k ⌋ + 1 colors. Proposition 4. Let G = (V, E) be a unit-disk graph, then the following inequality holds. χk (G) ≥
d(G) . 3k
Proof. Let H be the induced subgraph of G such that δ(H) = d(G). Consider a vertex v ∈ V (H) of minimum X-coordinate, and let H ′ denote the graph induced by N (v). Any proper co-k-plex coloring of G is also proper for H ′ , and it partitions H ′ into co-k-plexes. Hence we have, χk (G) ≥ χk (H ′ ) ≥
|V (H ′ )| d(G) ≥ . ′ αk (H ) 3k
Theorem 2. There exists a polynomial time 3-approximate algorithm for the minimum co-k-plex coloring problem in unit-disk graphs that uses at most 3χk (G) + 1 colors. Proof. Follows from Propositions 3 and 4.
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Conclusion
Generalizations of independent sets and graph coloring are studied in this paper in the context of unit-disk graphs. These are motivated by wireless network applications where the interference graph is modeled as a unit-disk graph, and the receivers are capable of distinguishing interference from a limited number of sources while extracting the desired signal. These relaxations called co-kplexes and co-k-plex coloring are known to be intractable on UDGs. We develop simple polynomial time algorithms that guarantee a constant performance factor for finding a maximum co-k-plex and for finding minimum co-k-plex coloring of UDGs. In the process, several classical results of Marathe et al. (25), Hochbaum (21), and Szekeres and Wilf (28) are extended for these relaxations and we positively settle a recent conjecture by Havet et al. (20) that the current best approximation ratio of co-k-plex coloring could be improved by a factor of two.
References [1] Andrews, J., Jacobson, M.: On a generalization of chromatic number. Congressus Numerantium 47, 33–48 (1985) [2] Balasundaram, B.: Graph theoretic generalizations of clique: Optimization and extensions. Ph.D. dissertation, Texas A&M University (2007)
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[3] Balasundaram, B., Butenko, S.: Optimization problems in unit-disk graphs. In: C.A. Floudas, P.M. Pardalos (eds.) Encyclopedia of Optimization. Springer Science + Business Media, New York (2008). To Appear. [4] Balasundaram, B., Butenko, S., Hicks, I.V.: Clique relaxations in social network analysis: The maximum k-plex problem. http://iem.okstate.edu/baski/files/kplex4web.pdf (2008). Submitted. [5] Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Computational Geometry: Theory and Applications 9(1-2), 3–24 (1998) [6] Cerioli, M.R., Faria, L., Ferreira, T.O., Protti, F.: On minimum clique partition and maximum independent set on unit disk graphs and penny graphs: complexity and approximation. In: Latin-American Conference on Combinatorics, Graphs and Applications, Electronic Notes in Discrete Mathematics, vol. 18, pp. 73–79 (electronic). Elsevier, Amsterdam (2004) [7] Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Mathematics 86, 165–177 (1990) [8] Cook, W., Cunningham, W., Pulleyblank, W., Schrijver, A.: Combinatorial Optimization. John Wiley and Sons, New York (1998) [9] Cowen, L., Cowen, R., Woodall, D.: Defective colorings of graphs in surfaces: partitions into subgraphs of bounded valence. Journal of Graph Theory 10, 187–195 (1986) [10] Cowen, L., Goddard, W., Jesurum, C.E.: Defective coloring revisited. Journal of Graph Theory 24(3), 205–219 (1997) [11] Dessmark, A., Jansen, K., Lingas, A.: The maximum k-dependent and f -dependent set problem. In: K.W. Ng, P. Raghavan, N.V. Balasubramanian, F.Y.L. Chin (eds.) Proceedings of the 4th International Symposium on Algorithms and Computation:ISAAC ’93, Lecture Notes in Computer Science, vol. 762, pp. 88–97. Springer, Berlin (1993) [12] Djidev, H., Garrido, O., Levcopoulos, C., Lingas, A.: On the maximum k-dependent set problem. Tech. Rep. LU-CS-TR:92-91, Dept. of Computer Science, Lund University, Sweden (1992) [13] Erlebach, T., Fiala, J.: Independence and coloring problems on intersection graphs of disks. In: E. Bampis, K. Jansen, C. Kenyon (eds.) Efficient Approximation and Online Algorithms, Lecture Notes in Computer Science, vol. 3484, pp. 135–155. Springer Verlag (2006) [14] Fishkin, A.V.: Disk graphs: A short survey. In: K. Jansen, R. Solis-Oba (eds.) Approximation and Online Algorithms, Lecture Notes in Computer Science, vol. 2909, pp. 260–264. Springer, Berlin (2004) [15] Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NPcompleteness. W.H. Freeman and Company, New York (1979) [16] Hale, W.K.: Frequency assignment: theory and applications. Proceedings of the IEEE 68(12), 1497–1514 (1980) [17] Harary, F., Jones, K.: Conditional colorability ii: Bipartite variations. Congressus Numerantium 50, 205–218 (1985) 8
[18] H˚ astad, J.: Clique is hard to approximate within n1−ǫ . Acta Mathematica 182, 105–142 (1999) [19] Havet, F., Kang, R.J., Sereni, J.S.: Improper colouring of unit disk graphs. Electronic Notes in Discrete Mathematics 22, 123–128 (2005) [20] Havet, F., Kang, R.J., Sereni, J.S.: Improper colouring of unit disk graphs. Tech. Rep. RR6206, Institute National de Recherche en Informatique et en Automatique (INRIA), France (2007). To appear in Networks. Available at: http://hal.inria.fr/inria-00150464 v2/ [21] Hochbaum, D.S.: Efficient bounds for the stable set, vertex cover and set packing problems. Discrete Applied Mathematics 6(3), 243–254 (1983) [22] Hunt, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. Journal of Algorithms 26(2), 238–274 (1998) [23] Kang, R.: Improper coloring of graphs. Ph.D. dissertation, University of Oxford (2007) [24] Krishna, P., Vaidya, N.H., Chatterjee, M., Pradhan, D.K.: A cluster-based approach for routing in dynamic networks. ACM SIGCOMM Computer Communication Review 27(2), 49–64 (1997) [25] Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995) [26] Ramaswami, R., Parhi, K.K.: Distributed scheduling of broadcasts in a radio network. In: Proceedings of the Eighth Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM ’89), vol. 2, pp. 497–504 (1989) [27] Seidman, S.B., Foster, B.L.: A graph theoretic generalization of the clique concept. Journal of Mathematical Sociology 6, 139–154 (1978) [28] Szekeres, G., Wilf, H.S.: An inequality for the chromatic number of a graph. Journal of Combinatorial Theory 4, 1–3 (1968)
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