Maximum Alliance-Free and Minimum Alliance-Cover Sets Khurram H. Shafique and Ronald D. Dutton School of Computer Science University of Central Florida Orlando, FL USA 3816 hurram@cs.ucf.edu, dutton@cs.ucf.edu Abstract A defensive alliance in a graph G = (V, E) is a set of vertices A V such that for every vertex v A, the number of neighbors v has in A is at least more than the number of neighbors it has in V A (where is the strength of defensive alliance). An offensive alliance is a set of vertices A V such that for every vertex v A, the number of neighbors v has in A is at least more than the number of neighbors it has in V A (where A is the boundary of set A and is defined as N[A] A). In this paper, we deal with two types of sets associated with these alliances: maximum alliance free and minimum alliance cover sets. Define a set X V to be maximum alliance free (for some type of alliance) if X does not contain any alliance (of that type) and is a largest such set. A set Y V is called minimum alliance cover (for some type of alliance) if Y contains at least one vertex from each alliance (of that type) and is a set of minimum cardinality satisfying this property. We present bounds on the cardinalities of maximum alliance free and minimum alliance cover sets and explore their inter-relation. The existence of forbidden subgraphs for graphs induced by these sets is also explored. Keywords: Alliance, Defensive Alliance, Offensive Alliance, Alliance free Set, Alliance Cover Set, Cohesive Set. 1 Definitions and Notations Alliances in graphs were first introduced by Hedetniemi, et. al.[3]. They proposed different types of alliances, namely (strong) defensive alliances, (strong) offensive alliances[1], global alliances[], etc. In this paper, we consider generalizations of offensive and defensive alliances which we refer to as alliances,
where the strength of an alliance is related to the value of parameter. Consider a graph G = (V, E) without loops or multiple edges. A vertex v in set A V is said to be satisfied with respect to A if deg A (v) deg V A (v) +, where deg A (v) = N(v) A = N A (v) = deg(v) deg V A (v). A set A is a defensive alliance if all vertices in A are satisfied with respect to A, where <. Note that a defensive ( 1) alliance is a defensive alliance (as defined in [3]), and a defensive 0 alliance is a strong defensive alliance or cohesive set [4]. Similarly, a set A V is an offensive alliance if v A, deg A (v) deg V A (v)+, where + <. Here, an offensive 1 alliance is an offensive alliance and an offensive alliance is a strong offensive alliance (as defined in [1]). A set X V is defensive alliance free ( daf) if for all defensive alliances A, A X Ø, i.e., X does not contain any defensive alliance as a subset. A defensive alliance free set X is maximal if v / X, S X such that S {v} is a defensive alliance. A maximum daf set is a maximal daf set of largest cardinality. Let φ (G) be the cardinality of a maximum daf set of graph G. For simplicity of notation, we will refer to a maximum daf set of G as a φ (G)-set. If a graph G does not have a defensive alliance (for some ), we say that φ (G)= V (G) = n, for example, φ (P n )= n, > 1. Since 1, a defensive alliance free set is also defensive 1 alliance free, we have φ 1 (G) φ (G) if and only if 1. We define a set Y V to be a defensive alliance cover ( dac) if for all defensive alliances A, A Y Ø, i.e., Y contains at least one vertex from each defensive alliance of G. A dac set Y is minimal if no proper subset of Y is a defensive alliance cover. A minimum dac set is a minimal cover of smallest cardinality. Let ζ (G) be the cardinality of a minimum dac set of graph G. Once again, we will refer to a minimum dac set of G as a ζ (G)-set. When G does not have a defensive alliance (for some ), we say that ζ (G)= 0. For offensive alliances, we define two types of alliance free (cover) sets depending on whether or not the boundary vertices of an offensive alliance affect the definition of the set. A set S V is offensive alliance free ( oaf) if for all offensive alliances A, A S Ø. S is wea offensive alliance free ( woaf) if for all offensive alliances A, N[A] S Ø. Similarly, a set T V is an offensive alliance cover ( oac) if for all offensive alliances A, A T Ø. T is a wea offensive alliance cover ( woac) if for all offensive alliances A, N[A] T Ø. The maximum (wea) offensive alliance free sets and minimum (wea) offensive alliance cover sets are defined in the same fashion as their defensive counterparts. For a graph G, we define the following invariants φ (G) = Size of a maximum daf set of G ζ (G) = Size of a minimum dac set of G
φ o (G) = Size of a maximum oaf set of G ζ o (G) = Size of a minimum oac set of G φ w (G) = Size of a maximum woaf set of G ζ w (G) = Size of a minimum woac set of G In this paper, we explore the properties and bounds of the above defined invariants and their relationship with each other. In general we will refer to both offensive and defensive alliances as alliances. Similarly, the terms alliance free set and alliance cover set will encompass all types of alliance free sets and cover sets defined in this section. For other graph terminology and notation, we follow [6]. Basic Properties Theorem 1. X V is a alliance cover if and only if V X is alliance free. Proof. A set X is a defensive alliance free set if and only if, for every defensive alliance A, A X Ø if and only if, for every defensive alliance A, A (V X) Ø if and only if V X is a defensive alliance cover. The justification for the (wea) offensive alliance cover is similar. Corollary. φ (G)+ζ (G) = φ o (G) + ζo (G) = φw (G)+ζw (G) = n Corollary 3. (i) If V is a minimal -dac ( oac) then, v V, there exists a defensive (offensive) alliance S v for which S v V = {v}. (ii) If V is a minimal -wdac then, v V, there exists an offensive alliance S v for which N[S v ] V = {v}. Since, 1 >, a alliance free set is also a 1 alliance free set and every 1 oaf set is a 1 woaf set, we have the following observation. Observation 4. For any graph G and < < 1, (i) 0 φ o (G) φ o 1 (G) φ w 1 (G) n (ii) 0 φ w 1 (G) φ w (G) n (iii) 0 φ (G) φ 1 (G) n
Also note that every daf set X is a woaf set. Suppose not, then there is an offensive alliance A such that N[A] X. Then v N[A], deg N[A] (v) deg V N[A] (v) +, which implies that N[A] is a defensive alliance and contradicts X being a daf set. Observation 5. φ w (G) φ (G) Suppose now a minimal 1 dac set Y, 1 > δ(g), and let A Y such that A is an offensive alliance. Let y A, then by Corollary 3, there exists a defensive 1 alliance S y such that S y Y = {y}. Hence x A Y such that deg A (x) deg V A (x) + 1. Also, since A is an offensive alliance, deg A (x) deg V A (x)+. Combining the two inequalities, we get, 1. This leads to the following observation: Observation 6. For any graph G and every 1, such that 1 > δ(g) and > 1, φ o (G) ζ 1 (G) 3 Defensive Alliance Free & Cover Sets For any, such that δ(g) < (G), we now that any independent set in a connected graph G is daf, therefore φ (G) β 0 (G), where β 0 (G) is the vertex independence number of graph G. We can further improve this bound by noting that the addition of any δ(g) + 1 vertices to an independent set will not produce a defensive alliance in the new set, hence, φ (G) β 0 (G) + δ(g) + 1. Since, every A V, such that A n δ(g) +, is a defensive alliance, φ (G) < n δ(g) +. Observation 7. If G is a connected graph and δ(g) < (G) then δ(g) δ(g) β 0 (G) + + 1 φ (G) < n + Next we present the values of φ (G) for some common graph families. Observation 8. If G is an Eulerian graph and δ(g) < i (G), then φ i 1 (G) = φ i (G). Observation 9. For the complete graph K n and n + 1 < < n, φ (K n ) = n + + for odd n for even n. n
Observation 10. For the complete bipartite graph K p,q, where p q and p < p, q + p + 1 for odd p φ (K p,q ) = q + p + 1 for even p. Note that the upper and lower bounds of Observation 7 coincide for both K n and K p,q, when is even. We have shown in [5] that the following lower bound holds for φ (G). Theorem 11. For every connected graph G and 0, n φ (G) +. We believe (but have been unable to prove) the following extension of the above theorem: Conjecture 1. If G is a connected graph and δ(g) < (G) then n φ (G) + Next, we show that no forbidden subgraph characterization exists for the graphs induced by minimal dac sets. Theorem 1. Let G be any graph and r an integer such that r. Then, for all r, there is a graph G, such that G contains G as an induced subgraph and ζ (G ) = r. Proof. Let a graph G = (V, E) where V = {v 1, v,..., v n } { and construct a graph G = (V, E ) as follows: V = V X Y, where X = x j i, 1 i n, 1 j max (r +, (G) + 1)} and Y } = {y 1, y,.{.., y r+ }. E = E E 1 E, where E 1 = {v i x j i, v i V, x j i X and E = x j i y l, x j i X, y l Y δ(g ) + 1, Thus δ(g ) = r + 1. Since by Observation 7, ζ (G ) we have ζ (G ) r+ 1 + 1 = r. Now consider C Y such that C = r. We claim that C is a dac set of graph G. Suppose not. Then there exists a defensive alliance S V C in G. Let v S. Since x X, deg(x) = r + 1, if v X then deg S (v) r + 1 < deg C (v) + = r +, which is contrary to S being a defensive alliance. Hence S X = Ø. Now let v V. By construction of graph G, v V, deg X (v) + (G) + 1 > deg V X(v) deg S (v), again a contradiction. The only remaining case is S Y, which is not possible as v S, deg S (v) = 0 < deg V S(y) + n(r + ) +. Hence S = Ø and C is a dac set. Thus ζ (G ) r. Combining the two results, we get ζ (G ) = r. }.
4 Offensive Alliance Free & Cover Sets In this section, we study the properties of the free sets and cover sets associated with offensive alliances. We begin by presenting the values of φ o (G) and (G) for some special classes of graphs. φ w Observation 13. For the complete graph K n, and n + 3 < < n n + φ o (K n ) = φ (K n ) 1 = 1 φ w (K n ) = n 1 Observation 14. For the complete bipartite graph K p,q, p q, and p + < q q + p + p & q both odd φ o (K p,q ) = q + p + p & q both even q + p + otherwise φ w (K p,q ) = n, p, q 1 It is interesting to note that while complete graphs attain the lower bound for φ (G), they have the maximum value for φ w (G). Lemma 15. If S is an offensive 1 alliance then (i) for all offensive alliances S V S such that 1 + > 0, S S = Ø. (ii) for all defensive alliances S V S such that 1 + > 0, S S = Ø. Theorem 16. For a connected graph G, if X is a maximal 1 woaf set and Y = V X then (i) > 1, Y is a woaf set (and hence, X is a woac set), and (ii) > max( 1, δ(g)), Y is a daf set (hence, X is a dac set). Proof. For i), let > 1 and suppose there exists an offensive alliance S for which N[S] Y. Let x S. From Corollary 3, there is an offensive 1 alliance S x for which N[S x ] Y = {x}. If x S x, then from Lemma 15, S and S x cannot be disjoint, a contradiction. So we must assume that x S x. But then, N(x) S x X, which leads to a contradiction since x must have at least one neighbor in S Y. Thus, Y is a woaf set and, from Theorem 1, X is a woac set.
For ii), let > max( 1, δ(g)) and suppose there exists a defensive alliance S Y. Let x S. From Corollary 3, there exists an offensive 1 alliance S x for which N[S x ] Y = {x}. If x S x then from Lemma 15, S and S x cannot be disjoint, a contradiction. So we must assume that x S x, but then N(x) S x X, which is not possible since deg S (x) (deg(x)+ )/ > 0. Hence, Y is a daf set and, from Theorem 1, X is a dac set. Corollary 17. (i) Every maximal 1 woaf set contains a minimal woac set, > 1. (ii) Every maximal 1 woaf set contains a minimal dac set, > max( 1, δ(g)). Since every woaf is also l woaf l >, by Theorem 1, every woac is also l woac. This observation leads to the following corollary of Theorem 16. Corollary 18. > 0, ζ w(g) n It is easy to prove that 0, ζ w (G) = n if and only if G K and <. We conclude this section by presenting a result for ζ w (G) similar to the one for ζ (G) in Theorem 1. Theorem 19. Let G be any graph and r an integer such that r 1. Then there is a graph G with ζ w (G ) = r, which contains G as an induced subgraph. Proof. Let a graph G = (V, E) where V = {v 1, v,..., v n } and construct a graph G = (V, E ) as follows: V = V X Y, where X = {x 1, x,..., x r } and Y is the union of disjoint sets Y 1, Y,..., Y r, such that i, Y i = n + 1. E = E E 1 E E 3, where E 1 = {v i x j, v i V, x j X}, E = r i=1 {x iy, y Y i } and E 3 = {yz y, z Y i, 1 i r}. Hence, G is obtained by adding r vertex disjoint cliques Y i {x i }, each of order n + vertices and maing each x i adjacent to every vertex of V. It is easy to see that X is a woac set of graph G, i.e. ζ w (G ) X = r. We claim that ζ w (G ) = r. Suppose not and let C V be a woac set of graph G such that C < r. By pigeon hole principle, there exists Y i such that (Y i {x i }) C = Ø. Since Y i = {x i } and deg Yi (x i ) = n++1 > deg V Y i (x i )+ = n +, Y i is an offensive alliance in G such that N[Y i ] V C, which is contrary to C being a woac set of graph G. Hence ζ w (G ) r. Combining the two results, we get ζ w (G ) = r. 5 Open Problems 1. Determine the computational complexity of finding each of φ (G), φ o (G) and φ w (G).
. Find efficient algorithms for computing φ (G), φ o (G) and φw (G). 3. Determine tight upper and lower bounds for φ o (G) and φw (G) and characterize extremal graphs. 4. Determine the values of φ (G), φ o (G) and φw (G) for other classes of graphs, for example, grid graphs. 5. Do results similar to Theorem 1 and Theorem 19 hold for ζ o (G)? 6. Study the cover and free sets of other alliances, e.g., dual alliances and global alliances, and their relationship. References [1] O. Favaron, G. Frice, W. Goddard, S. M. Hedetniemi, S. T. Hedetniemi, P. Kristiansen, R. C. Lasar, and D. Saggs, Offensive alliances in graphs. Preprint, 00. [] T. W. Haynes, S. T. Hedetniemi, and M. A. Henning, Global defensive alliances in graphs. Preprint, 00. [3] S. M. Hedetniemi, S. T. Hedetniemi, and P. Kristiansen, Alliances in graphs. Preprint, 000. [4] K. H. Shafique and R. D. Dutton, On satisfactory partitioning of graphs, Congressus Numerantium 154 (00), pp. 183-194. [5] K. H. Shafique and R. D. Dutton, A tight bound on the cardinalities of maximum alliance-free and minimum alliance-cover sets. Preprint, 00. [6] D. B. West, Introduction to Graph Theory. Prentice Hall, NJ, 001.