On Immunity and Catastrophic Indices of Graphs

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1 On Immunity and Catastrophic Indices of Graphs RASTISLAV KRÁLOVIČ Comenius University, Slovakia PETER RUŽIČKA Comenius University, Slovakia Abstract Immunity index of a graph is the least integer c 1 such that each configuration of size c 1 is immune. Catastrophic index of a graph is the least integer c such that each configuration of size c is catastrophic. This paper contains the first systematic study of immunity indices on a variety of interconnection networks and their distance from catastrophic indices. Keywords distributed computing, faul tolerance, interconnection networks 1 Introduction We consider the following coloring game played on a simple connected graph. The game proceeds in synchronous rounds and uses a set of colors black white. Initially, vertices are colored black or white. At each round, each white vertex simultaneously recolors itself by the color of the simple majority of its neighbours. A set of vertices M is said to be a dynamic monopoly (shortly dynamo) ifstarting with only the vertices of M colored black the game eventually reaches the all-black configuration. Such a dynamo is called irreversible to stress the fact that initially black vertices are permanently black (i.e. they cannot be recolored to white). The importance of this game follows from the fact that it models faulty behaviour of point-to-point systems based on majority voting. In this case, dynamos correspond to sets of initial faults that cause the entire system to fail. An immune subgraph of a graph is a subgraph for which there does not exist a computation starting from some initial configuration with all vertices of the immune subgraph white and ending in all-black configuration. In this case, immune Supported in part by grant from VEGA 1/7155/0. 31

2 3 Sirocco 8 subgraphs correspond to sets of correct nodes that prevent the entire system from falling. Dynamic monopolies have been introduced in [L+93] and since that time they have been intensively studied in the literature. The main line of the previous research has been oriented on the size of dynamos for various topologies (see e.g. survey paper [P01]). Recently, the systematic study of dynamos with respect to both the size and the time (i.e. how many rounds are needed to reach the all-black configuration) was presented in [F+00]. Immunity subgraphs has been introduced in [P96], where the connection between immunity and expansion has been studied. In the dynamo size problem we ask whether there is a catastrophic configuration of certain size. The natural question to ask is: For which sizes are all configurations catastrophic. This is formalized in the notion of catastrophic index. In analogy, we can introduce the notion of immunity index. We define immunity index c 1 of a graph G as the least integer such that each configuration of size c 1 is immune. Clearly, c 1 N m 1, where m is the size of a smallest dynamo of N-vertex graph G. We also introduce catastrophic index c of a graph G as the least integer such that each configuration of size c is catastrophic. Later on it is proved that c N d 1, where d is the size of a minimum immune subgraph of N-vertex graph G. In this paper we propose to study catastrophic indices of graphs, i.e. to study minimal immune subgraphs in detail. We are interested in the following questions: (1) What is the size of smallest immune subgraphs in a given graph? () Given a graph, can we partition it into (minimal) immune subgraphs? ( The maximal number of disjoint immune subgraphs in such a partition is called the immunity degree of G. ) Basic Notions Let G V Eµ be a simple connected graph and N V. Assume that the vertices of the graph G are colored (black or white). By a configuration on G we mean a partition of V to the set of black and the set of white vertices. For simplicity, a configuration will be referred to as the set of its black vertices. By the size of a configuration we mean the number of its black vertices. The final configuration (all-black configuration) of G is the configuration with all vertices black. A catastrophic configuration is a configuration from which there is a synchronous computation (i.e. each round of which is performed by means of a simple majority rule applied to each white vertex of G simultanously) leading to the all-black configuration. Such a computation is called a complete computation.

3 Královiˇc &Ruˇziˇcka: Immunity and Catastrophic Indices 33 By time complexity of a complete computation we mean the number of its rounds. A dynamic monopoly (shortly dynamo) is a set of black nodes whose corresponding configuration is catastrophic. By t-time computation we mean a complete computation having t rounds. By t-time dynamo we mean a dynamo having complete computation with t rounds. If not mentioned explicitly, by a dynamo we mean an irreversible dynamo using simple majority rule. By a minimal dynamo we mean a dynamo of minimal size. Given a dynamo M,letM t denote the set of black-colored vertices after round t (with M 0 M). A dynamo M is monotone if M t M t 1 for every t 0. It is obvious that irreversible dynamos are monotone. An immune subgraph G ¼ of G is a subgraph for which there does not exist a complete computation on G starting from some initial configuration with all vertices from G ¼ white. This means that a subgraph G ¼ G is immune if and only if for each vertex v ¾ V ¼ it holds deg G ¼ vµ Neigh G vµ V ¼ deg G vµ.by t-immune subgraph we mean an immune subgraph G ¼ G such that there does not exist a t-time computation on G starting from a configurationwith all vertices from G ¼ white. An immune configuration is a set of vertices of some immune subgraph of G. By catastrophic index of a graph G we mean the least integer c such that each configuration with at least c black vertices is catastrophic. By immunity index of a graph G we mean the least integer c such that each configuration of size c is immune. The immunity degree is the maximal number of disjoint immune subgraphs in G. 3 Catastrophic indices and smallest immune subgraphs Recently, there has been intensive study on the size of dynamos for various topologies. Tight size bounds have been obtained for complete trees, rings [F+00], tori [F+98], butterflies [LPS99, F+00], cube-connected cycles [LPS99] and shuffleexchange graphs [F+00]. There are still asymptotical differences between the upper and lower bounds on the size of minimal dynamos for wrap-around butterflies, De Bruijn graphs, hypercubes and star graphs and the open question is to tighten these differences. In the problem of dynamo size we ask about the existence of catastrophic configuration of certain size. A natural question is to ask, for which sizes all configurations are catastrophic. Next property relates the catastrophic index of graphs to the size of their smallest immune subgraphs. Theorem 1 For the catastrophic index c of G it holds c N m 1,whereNis the number of vertices in G and m is the size of the smallest immune subgraph of G.

4 34 Sirocco 8 Proof. Let G ¼ be the minimal immune subgraph of G with m vertices. Clearly, for each c ¼ N m there is a configuration with c ¼ black vertices which is not a dynamo (the configuration with G ¼ white), hence c N m. Consider a configuration with N m 1 black vertices. Suppose, for the sake of contradiction, that it is not a dynamo. Then the computation stops in a configuration with at most m 1 white vertices. Each of the white vertices has more that half of its neighbours white, hence there is an immune subgraph of size m 1. ¾ As a consequence of the previous theorem, several interesting estimates can be obtained. Observation 1 For each N 3 there exists an N-vertex graph with the catastrophic index at least N 3. Proof. Consider a graph K 4 sharing one vertex with an arbitrary N 4µ-vertex graph. Then K 3 is an immune subgraph. ¾ Observation The catastrophic index of an r-regular graph G is at most N r 1, where N is the number of vertices in G. Proof. Follows from the fact that each immune subgraph of an r regular graph G has at least r vertices. ¾ 4 Size of minimal immune subgraphs As we have seen in the previous subsection, the crucial point in determining the catastrophic indices of graphs is to determine the size of the smallest immune subgraphs. The first question to ask is about the complexity of the following decision problem: Given a graph G andanintegerk, is there an immune subgraph of G of size k? In the next theorem we show that this problem is NP-complete. Theorem The following decision problem is NP-complete: Given a graph G and an integer k, is there an immune subgraph of size at most k? Proof. Reduction from EXACTCOVERBY3-SETS (EC3S). The instance of EC3S consists of a set A a 1 a 3m and a system of subsets of A:S S 1 S n such that S i A, S i 3. The instance is positive, if there exists a setc C 1 C m such that C i ¾S and m i 1 C i A. Construct a graph G V Eµ as follows. Start with a complete bipartite graph with bipartition v 1 v n and a 1 a 3m. For each set S i add vertices b i1 b i b i3 with degree two, such that b ij is connected with v i and a i j,where

5 Královiˇc &Ruˇziˇcka: Immunity and Catastrophic Indices 35 S i a i1 a i a i3. Moreover, let us call a stop-tree the graph consisting of a root and 15 m nµ leaves. For each vertex v i add 3m stop-trees with roots connected to it, and for each vertex a i add m stop-trees with roots connected to it. It holds that G has an immune subgraph of size at most 7m if and only if the corresponding instance of EC3S is positive. Clearly if the EC3S instance is positive, consider the subgraph I G consisting of all vertices a i, vertices v j such that S j ¾C,and vertices b kl such that v k ¾ I. I is immune and has size 7m. On the other hand, if there exist an immune subgraph of size 7m, it contains no vertices from stop-trees. Moreover, it contains all vertices a i.fromthesize requirements it follows that the vertices v i contained in I form a positive instance of EC3S. ¾ There are similar problems to the problem of minimal immune subgraphs, namely 1-step irreversible (reversible) monopoly: reversible monopoly M R : v ¾ G Γ vµ M R deg vµ irreversible monopoly M I : v ¾ G M I Γ vµ M I deg vµ immune subgraph I : v ¾ I Γ vµ I deg vµ where Γ vµ u ¾ V Gµ d G u vµ 1 Another question to ask is: what is the size of minimal immune subgraphs in well-known interconnection networks? We give tight bounds for tori, cubeconnected cycles and hypercubes, and upper bounds for butterflies and star networks. In the proof of the following theorem we need this useful lemma. Lemma 1 [Ch+88] Edge isoperimetric inequality for a d-dimensional hypercube Q d is Qd mµ m d log mµ where G mµ denotes the size of the minimal edge boundary of an m vertex subset of vertices of G. Theorem 3 The size of a minimal immune subgraph in (1) a tori T d1 d is min d 1 d µ; () a hypercube Q d is d 1 ; (3) a butterfly BF d is at most 3 d 1µ 4 d mod µ 8; (4) a cube connected cycles CCC d,d 8,is8; (5) a star S d is at most d 1 µ!.

6 36 Sirocco 8 Proof. (1) W.l.o.g. let d 1 d. Consider an orientation such that the up-down direction is in the dimension of d 1. First we show that if S is a minimal immune subgraph, then S contains one full column of vertices. From that, the result follows: every vertex in S has at least three neighboursin S so it is possible to assign to each vertex in the row of d 1 vertices a distinct vertex in S. To show that there is one row in S first note that two rows form an immune subgraph of d 1 vertices. Consider an arbitrary immune subgraph S. As every vertex in S has at least three neighbours in S there is a closed path p consisting of down and right edges only. Suppose there is no such column. Then there are two possibilities for p. Ifp is a complete row then by argument similar to above S contains d vertices and is not minimal. If p is not a complete row then it must cross all four borders hence the length is at least d 1 d and S is not minimal. () Let S be an immune subgraph of Q d, S m. For each vertex v ¾ S there are d 1 neighbours in S, so there are at most m d 1 edges outgoing from m d log mµ which implies the claim. S. Using Lemma 1 we get m d 1 (3) Let k d. Consider vertices 0d kµ and 0 k 1 d k kµ together with complete binary trees connecting each of them to the set 0 k α d 1µ α ¾ 0 1 d k and vertices 1 d kµ, and 1 k 0 d k kµ together with complete binary trees connecting each of them to the set 1 k α d 1µ α ¾ 0 1 d k. Similarly consider vertices 0 d k 1µ and 0 k 1 d k k 1µ together with complete binary trees connecting each of them to the set 0 k α 0µ α ¾ 0 1 d k and vertices 1 d k 1µ, and 1 k 0 d k k 1µ together with complete binary trees connecting each of them to the set 1 k α 0µ α ¾ 0 1 d k. This subgraph is immune and contains four trees of height d 1µ and four trees of height d 1µ, where each leaf is in exactly two trees. Thus the number of vertices is 3 d 1µ 4 d mod µ 8. (4) CCC is a cubic graph, hence its minimal immune subgraph is the shortest cycle. (5) Star S d can be decomposed into d substars S d 1 by fixing each different symbol in 1 d in one particular position to d. If we fix a specific symbol in the last position we observe that there are d 1µ! vertices (i.e. an S d 1 )for every one of the d symbols. Thus, the vertices of the S d can be partitioned into d groups, each containing d 1µ! vertices and each being isomorphic to S d 1. If this decomposition is recursively applied to the resulting substars, S d can be decomposed into d! k! substars S k,1 k d 1. ¾ N d 1, Note that following [F+00] it holds for the case of star graph that m where m is the size of a minimal dynamo on S d. We are aware of only trivial upper bound on the size of a minimal dynamo in the form N fors d.

7 Královiˇc &Ruˇziˇcka: Immunity and Catastrophic Indices 37 5 Decomposition to immune subgraphs In the previous subsection we have concentrated on the problem of determining the size of minimal immune subgraphs. To recall, our motivation to study the minimal immune subgraphs was that they correspond to the failure patterns that does not cause the entire system to fail. Another interesting aspect of this problem is to know how many disjoint immune subgraphs are there. This, in certain sense expresses the immunity degree of the entire distributed system. In this subsection we give characterization results of the following type: Given a graph G, is there a unique (up to some symmetries) decomposition to minimal immune subgraphs? We give tight immunity degree values for tori, cubeconnected cycles and hypercubes, and give lower bounds for butterflies and star graphs. Theorem 4 There is a decomposition of (1) a tori T d1 d,d 1 d,d even, to minimal immune subgraphs; () a butterfly BF d to immune subgraphs; (3) a cube connected cycles CCC d, d even, to minimal immune subgraphs; (4) a hypercube Q d to minimal immune subgraphs; (5) a star S d to immune subgraphs; Proof. The part (1) is trivial, () follows from [LPS99], (4) and (5) follow from the recursive structure of these graphs and the previous theorem. For the part (3) consider the faces of the original hypercube (i.e. cycles of length 8 in CCC) corresponding to dimensions i i 1µ for 0 i d. ¾ As a consequence, we obtain exact immunity degree values for certain topologies. The immunity degree of a tori T d1 d is d,(d 1 d ). The immunity degree of CCC d, d even, d 8, is d d 3. The immunity degree of Q d is d 1.Wealso have lower bounds on immunity degrees for other topologies. Immunity degree of BF d is at least d 1 and of S d is at least d d 1µ d 1 1µ. Immunity degree of a graph gives a lower bound on the size of a minimal dynamo in this graph. For several topologies we obtained rather weak lower bounds. However, for CCC d (and also wrapped butterflies but the result is not mentioned here) we get the best known lower bounds, previously established in [LPS99]. 6 Minimal t immune subgraphs Up to now, we have concentrated on the size of dynamos and immune subgraphs. The size of dynamo is clearly a crucial parameter: a large size implies a less

8 38 Sirocco 8 likely occurence. Thus, a system in which the smallest dynamo is large has a high degree of fault-tolerance. However, the size is not the only interesting aspect of the quality of dynamos. In particular, the time needed for a dynamo to converge into all-black configuration is a very important characteristic, not only from a combinatorial but also from a practical point of view. If a catastrophic set of faults has a slow evolution, its presence might be more easily detected; and on the other hand, a fast dynamo is inherently more dangerous for the system. The systematic study of dynamos with respect to both the size and the time has been done in [F+00] for various models and topologies. Tight tradeoffs between the size and the time has been presented for rings, complete d-ary trees, tori, wraparound butterflies, cube-connected cycles and hypercubes. The question remains to determine the size of minimal t-time irreversible dynamos for De Bruijn graphs. The study of minimal t-immune subgraphs is important, mainly for their close relationship to the problem of determining t-catastrophic indices of networks. We give tight bounds on the size of minimal t-time immune subgraphs for rings, tori and hypercubes and upper bounds for butterflies. We define t-immunity index of a graph G as the smallest integer c such that each configuration of size c is t-immune. Clearly, c N m 1 wherem is the size of a minimal t-dynamo of N-vertex graph G. Wealsodefinet-catastrophic index of a graph G as the smallest integer c such that each configuration of size c is t-catastrophic. Clearly, c N d 1whered is the size of a minimum t-immune subgraph of N-vertex graph G. Let B G v rµ denote a ball with radius r in graph G centered in vertex v (i.e. the set of vertices in G with distance at most r from v). Note, that while every ball with radius t of an immune subgraph is t-immune, it may be the case that a t-immune subgraph is not contained in any minimal immune subgraph. As an example consider a graph G consisting of two cliques of size t and one hypercube of dimension t, each connected to a root vertex v with paths of length t. Aball centered in v with radius t is t-immune subgraph of G with 1 3t vertices. We claim that each minimal immune subgraph of G is fully contained in the hypercube. First note that any t -dimensional subcube (not containing the vertex on the path towards v) is immune subgraph of G with t vertices. On the other hand every immune subgraph containing v contains at least one vertex from a t -clique and hence its size is at least than t 1. Theorem 5 Minimal t-immune subgraphs of (1) N-vertex ring R N are of size t 1 (1 t N 1 ); () square n n tori are of size 4t (1 t N 1 ); (3) d-dimensional butterfly BF d areofsizeatmost3 t for (1 t (4) d-dimensional hypercube Q d are of size t i 0 d 1 i ; d 1 );

9 Královiˇc &Ruˇziˇcka: Immunity and Catastrophic Indices 39 Proof. (1) The minimal immune subgraph of a ring R N is of size N. Fort, 1 t N 1, any consecutive subset of t 1 vertices forms a t-immune subgraph. Aseverycomputationstarting froma configurationwith atmostt white vertices has time at most t 1, this t-immune subgraph is minimal. () Consider two neighbouring columns in tori T n n. It is obvious that they form minimal immune subgraph T of T n n.asb T v tµ has size 4t we have the upper bound. For the lower bound note that every configuration with more than four and less than n 1 vertices has at least four corners (by a corner we mean a black vertex with at most two black neighbours) and each configuration with four or n 1 vertices has at least three corners. Consider a computation starting from a configuration with fewer that 4t black vertices. The lower bound comes from the fact that as t N 1 in each step at least four vertices will be recolored black. (3) Let k d. Consider BT 0 0 kµ tµ where T is the immune subgraph from Theorem 3. For t k it consists of two complete binary trees of heights t and t 1 thus having t 1 1 t 1 vertices. (4) Let k d 1. From Theorem 3 it follows that the minimal immune subgraph is the k-dimensional subcube. The upper bound is obtained by B Qk v tµ for any vertex v. For the lower bound we prove the following: for any t-immune subgraph G ¼ k there is a vertex v such that there are at least i vertices in G¼ that have distance i from v. Consider a computation starting from a configuration with only vertices from G ¼ white. Clearly, as G ¼ is t-immune there must be a vertex v that is white after t steps. Let S 0, S 1,..., S t be a sequence of sets t i 0 S i G ¼, where S t i are vertices that are recolored after i 1steps(v¾S 0 ). Each w ¾ S i, 0 i t must have at least k neighbours in i 1 j 0 S j. Moreover, w has at most distance i from v, hence at most i neighbours in i j 0 S j with distance i 1 from v. It follows that v has at least k i neighbours W in S i 1 with distance i 1 from v. Each of these can have at most t 1 neighbours in S i. We use induction to argue that there is at least k i vertices in Si and hence at least k i 1 vertices in Si 1. ¾ We have only trivial upper bound on the size of minimal t-immune subgraph in star graphs. The question remains to determine exact bounds. 7 t dominating set of minimal immune subgraphs Decomposition of a network to minimal t immune subgraphs with respect to perfect t dominating sets is important in order to guarantee the fault-tolerant quality of t time simple majority computations in networks. In the next theorem we give the tight size of minimal t-dominating set (i.e. dominating set with radius t) of a minimal immune subgraph for a squared tori and an upper bound on the size for butterflies. The question for hypercubes and star graphs seems to be far from trivial.

10 40 Sirocco 8 Theorem 6 Minimal t dominating set of a minimal immune subgraph of (1) squared tori T n n is of size t n (1 t n 1 ); () BF d is of size at most d 1 d 1 t 1, 1 t, t 1 d 1. Proof. (1) Let T be the minimal immune subgraph of tori T n n. Consider B T vµ as in the proof of the previous observation. The size of B T vµ is 4t for 1 t N 1.Thereare N t disjoint BT vµs in each T. The distance of its centers is t 1. It follows that the set of centers of B T vµs forms minimal t-dominating set of a minimal immune subgraph of T n n. () Follows from immune subgraphs constructed in [LPS99] and from directly constructed t-dominating sets of butterfly columns of size d 1. ¾ In the next observation we present partial results for 1-dominating sets. Observation 3 (1) Minimal 1 dominating set of a smallest immune subgraph of BF d is of size at most d 1 d 1 3. () Minimal perfect 1 dominating set of a smallest immune subgraph of Q d is d 1 of size for d d 1 1 k 1µ. (3) Minimal 1 dominating set of a smallest immune subgraph of S d, d even, is of size at most 1 3 d µ!. Proof. (1) The consequence of the previous theorem for t 1. () The result appears e.g. in [DR97]. (3) Perfect 1 dominating set of S 3 is of size. There is a decomposition of d µ! S d 1 into 3! substars S 3, thus the result follows. ¾ 8 Distances of immunity and catastrophic indices In this section we give some notes concerning the relationship between the sizes of immunity and catastrophic indices. First,we show that there are graphs having immune subgraphs of size Θ nµ. The trivial example is a ring R N, but there are also non-constant-degree graphs with this property. As an example of k-regular graphs with large minimal immune subgraphs we present the class of Ramanujan graphs [LPS88]: k-regular Cayley graphs with the second smallest eigenvalue of their Laplacian matrix bounded by λ k Ô k 1. These graphs satisfy a number of extremal combinatorial properties (e.g. large girth). Ramanujan graphs are also the best known explicit expanders.

11 Královiˇc &Ruˇziˇcka: Immunity and Catastrophic Indices 41 Theorem 7 Ramanujan graphs have immune subgraphs of size Θ Nµ. Proof. Let λ be the second smallest eigenvalue of Laplacian. Consider a vector x of vertex weights, such that x v 1 for vertices from the immune subgraph and x v 1 otherwise. Following [F75] it holds N v i v j µ¾e x i x j µ λ vi v j ¾V x i x j µ Consider a Ramanujan graph of degree k.letδ k 1 As λ k Ô k 1, it follows that I N 1 k 1 k 4 Ô k 1. It follows that λ Nδ N I. ¾ Another interesting question is to characterize graphs, for which the immunity and catastrophic indices are equal (or close); of great difference. We mention some notes on special topologies where immunity and catastrophic indices are close (tori T d1 d for d d 1 ) or of great differences (rings). Ring R N : immunity index is c 1 N; catastrophic index is c 1; immunity degree is 1. Tori T d1 d : immunity index is c 1 d 1 d d 1; catastrophic index is c d 1 d d 1 1; immunity degree is d. 9 Conclusions We have initiated the study of immune subgraphs with respect to both the size and time size complexities. We have shown that to determine whether there is an immune subgraph of given size is NP-complete in general graphs. We have derived tight sizes of minimal immune subgraphs for tori, cube-connected cycles and hypercubes and tight time size tradeoffs for rings, tori and hypercubes. Further results on immunity degrees and t-dominating sets of minimal immune subgraphs are also given. 10 Acknowledgements This paper has benefited from many discussions the authors have had with Shmuel Zaks during his stay at Comenius University.

12 4 Sirocco 8 References [Ch+88] F.R.K. CHUNG, Z.FÜREDI, R.L. GRAHAM, P.D.SEYMOUR: On Induced Subgraphs of the Cube. Journal of Combinatorial Theory A-49: (1988) [DR97] S. DOBREV, P.RUŽIČKA: Linear Broadcasting and N loglogn Election in Unoriented Hypercubes. Proc. 4th Colloq. on Structural Information & Communication Complexity (SIROCCO), Carleton Scientific, Ottawa, (1997) [F75] M. FIEDLER: A property of eigenvalues of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Mathamatical Journal 5: (1975) [F+98] P. FLOCCHINI, E.LODI, F.LUCCIO, L.PAGLI, N.SANTORO: Irreversible Dynamos in Tori. Proc. Euro-Par, Southampton, England, 1998, [F+00] P. FLOCCHINI, R.KRÁLOVIČ, A.RONCATO, P.RUŽIČKA, N.SAN- TORO: On Time versus Size for Monotone Dynamic Monopolies in Regular Topologies. Proc. 7th Colloq. on Structural Information & Communication Complexity (SIROCCO), L Aquilla, Italy, June 000, (Full version will appear in the Journal of Discrete Algorithms, 001) [L+93] N. LINIAL, D.PELEG,Y.RABINOVICH, M.SAKS: Sphere packing and local majorities in graphs. Proc. nd ISTCS, IEEE Computer Soc. Press, June 1993, [LPS88] A. LUBOTZKY, R.PHILLIPS, P.SARNAK: Ramanujan Graphs. Combinatorica 8(3): 61 77, (1988) [LPS99] F. LUCCIO, L.PAGLI, H.SANOSSIAN: Irreversible Dynamos in Butterflies. Proc. 6th Colloq. on Structural Information & Communication Complexity (SIROCCO), Bordeaux, 1999, Carleton University Press [P96] D. PELEG: Graph Immunity Against Local Influence. Tech. Report CS96-11, The Weizmann Institute, 1996 [P01] D. PELEG: Local Majorities, Coalitions and Monopolies in Graphs: A Review. Theoretical Computer Science, to appear Rastislav Královič is with the Department of Computer Science, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia. kralovic@dcs.fmph.uniba.sk Peter Ružička is with the Institute of Informatics, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia. ruzicka@dcs.fmph.uniba.sk