Secure sets and defensive alliances faster algorithm and improved bounds. Amano, Kazuyuki; Oo, Kyaw May; Otach Author(s) Uehara, Ryuhei

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1 JAIST Reposi Title Secure sets an efensive alliances faster algorithm an improve bouns Amano, Kazuyuki; Oo, Kyaw May; Otach Author(s Uehara, Ryuhei Citation IEICE Transactions on Information an E98-D(3: Issue Date Type Journal Article Text version publisher URL Rights Copyright (C2015 IEICE. Kazuyuki Am Oo, Yota Otachi, an Ryuhei Uehara, Transactions on Information an Syst D(3, 2015, Description Japan Avance Institute of Science an

2 VOL. E98-D NO. 3 MARCH 2015 The usage of this PDF file must comply with the IEICE Provisions on Copyright. The author(s can istribute this PDF file for research an eucational (nonprofit purposes only. Distribution by anyone other than the author(s is prohibite.

3 486 IEICE TRANS. INF. & SYST., VOL.E98 D, NO.3 MARCH 2015 PAPER Special Section on Founations of Computer Science New Spirits in Theory of Computation an Algorithm Secure Sets an Defensive Alliances in Graphs: A Faster Algorithm an Improve Bouns Kazuyuki AMANO a, Member, Kyaw May OO b, Nonmember, Yota OTACHI c, an Ryuhei UEHARA, Members SUMMARY Secure sets an efensive alliances in graphs are stuie. They are sets of vertices that are safe in some senses. In this paper, we first present a fixe-parameter algorithm for fining a small secure set, whose running time is much faster than the previously known one. We then present improve boun on the smallest sizes of efensive alliances an secure sets for hypercubes. These results settle some open problems pause recently. key wors: secure set, efensive alliance, fixe-parameter tractability, hypercube 1. Introuction The concept of a efensive alliance in a graph is introuce by Kristiansen, Heetniemi, an Heetniemi [14]. Intuitively, a efensive alliance is a set of vertices that is safe from attacks by its neighborhoo. Later the concept of a secure set in a graph is introuce by Brigham, Dutton, an Heetniemi [1]. Roughly speaking, a secure set is a much safer set of vertices than a efensive alliance. As the reaers see from the formal efinitions in the next section, any secure set is a efensive alliance. The problem of fining a smallest secure set or a smallest efensive alliance is of interest. These concepts have been intensively stuie (see e.g. [1], [4], [5], [10], [14]. Recently, Isaak, Johnson, an Petrie [12], [16] have introuce fractional variants of secure sets. Jamieson, Heetniemi, an McRae [13] showe that the problem of eciing whether a graph has a efensive alliance of size at most given k is NP-complete. Fernau an Raible [9] showe that the problem is fixe-parameter tractable when parameterize by the size of a efensive alliance. Enciso an Dutton [6] later presente a fixeparameter algorithm with a better running time. To the best of our knowlege, the complexity of fining a smallest secure set is not known. It is known that given a Manuscript receive February 28, Manuscript revise June 28, The author is with the Depertment of Computer Science, Gunma University, Kiryu-shi, Japan. The author is with University of Computer Stuies, Yangon, No. 14/20, Than Lwin Roa, Bahan, Yangon, Myanmar. The authors are with School of Information Science, Japan Avance Institute of Science an Technology, Nomi-shi, Japan. a amano@cs.gunma-u.ac.jp b kmayoo19@gmail.com c otachi@jaist.ac.jp uehara@jaist.ac.jp DOI: /transinf.2014FCP0007 graph an a set of vertices, it is conp-complete to ecie whether the set is a secure set of the graph [3]. Enciso an Dutton [7] presente a fixe-parameter algorithm for solving the problem of eciing whether a given n-vertex graph has a secure set of size at most k in time O(2 k log 2k n. 1.1 Our Results In Sect. 3, we present a faster fixe-parameter algorithm with a running time of O(2 3k k 2 n for fining a secure set of size at most k. In Sect. 4, we present graph-theoretic contributions. For hypercubes, we first etermine exactly the efensive alliance number, an also the efensive alliance partition number as a corollary. We then generalize these results to Hamming graphs. Next we present a significantly improve lower boun of the security number of hypercubes. The previously known lower boun ue to Petrie [16] is 2 /2, an our lower boun is more than for large enough. 2. Preliminaries All graphs in this paper are finite, simple, an unirecte. For a graph G, we enote its vertex set an ege set by V(G an E(G, respectively. The (open neighborhoo N(v of a vertex v V(G istheset{u {u,v} E(G}. The egree of v V(G is N(v an enote by eg G (v. The close neighborhoo N[v]ofavertexv V(Gistheset{v} N(v. For a subset S V(G, we efine its close neighborhoo as N[S ] = v S N[v]. The subgraph inuce by S V(G is enote by G[S ]. The istance between u,v V(G, enote ist(u,v, is the length of a shortest u v path in G. The Cartesian prouct of graphs G an H, enote G H, is the graph with the vertex set V(G V(H an the ege set {{(g, h, (g, h } g V(G, {h, h } E(H} {{(g, h, (g, h} h V(H, {g, g } E(G}. Theth Cartesian power of a graph G, enote G, is efine as G 1 = G an G = G G 1 for 2. The Hamming graph K k is the th Cartesian power of the complete graph K k for some an k. The-imensional hypercube Q is the th Cartesian power of K 2 ; that is, Q = K 2. Let G be a graph. A non-empty set S V(G isa efensive alliance if for each s S, N[s] S N[s] \ S. The efensive alliance number a(g ofg is the size of a smallest efensive alliance of G. A non-empty set S V(G is a secure set if for each X S, N[X] S N[X]\S (see Copyright c 2015 The Institute of Electronics, Information an Communication Engineers

4 AMANO et al.: SECURE SETS AND DEFENSIVE ALLIANCES 487 [1]. The security number s(g ofg is the size of a smallest secure set of G. Clearly, a(g s(g for any graph G. 3. A Faster Fixe-Parameter Algorithm for Secure Sets Recall that the running time of the algorithm by Enciso an Dutton [7] is O(2 k log 2k n, where n is the number of vertices of G. Here we present an improve O(2 3k k 2 n-time algorithm. Theorem 3.1. The problem of eciing whether s(g k can be solve in time O(2 3k k 2 n. Proof. For any S V(G, the following claims follows from the efinitions (see [7]: Claim 3.2. If N[S ] > 2k, then there is no secure set S that satisfies S S an S k. Claim 3.3. If S S is a minimal secure set, then there is a vertex u S \ S such that u N[S ] \ S. Now we escribe our algorithm. See Algorithm 1. We first guess a vertex v V(G, an call Fin({v}, to fin a secure set containing v. ForS V(G an F N[S ]\S,the proceure Fin(S, F fins a secure set S of size at most k such that S S an F S =. IfS itself is secure, then Fin(S, F outputs S an stops. Assume S is not secure an there is a minimal secure set S S with S kan S F =. Clearly, S < k. Since F N[S ] \ S,wehave F S k. By Claim 3.2, N[S ] 2k. Hence, if not all the conitions are satisfie, then there is no such S. If all the conitions are satisfie, the proceure recursively fins such S. If S exists, then there is a vertex u S \ S such that u N[S ] \ (S F by Claim 3.3. Moreover, if u N[S ] \ (S F, then either u S \ S or u N[S ] \ S hols. We a u into S in the former case an into F in the latter case. To check both cases, we call Fin(S {u}, F an Fin(S, F {u}. Algorithm 1 Fin a secure set of G with size at most k. 1: for all v V(G o 2: Fin({v}, 3: proceure Fin(S, F 4: if Secure(S then 5: Output S, an stop. 6: if S < k an F kan N[S ] 2kthen 7: u := a vertex in N[S ] \ (S F 8: Fin(S {u}, F 9: Fin(S, F {u} 10: proceure Secure(S 11: for all X S o 12: if N[X] S < N[X] \ S then 13: return false 14: return true Next we show the running time. We first have n branches corresponing to the calls of Fin({v}, for all v V(G. Each call of Fin(S, F has no or two branches that correspon to the calls of Fin(S {u}, F an Fin(S, F {u} if S < k, F k, an N[S ] 2k. Therefore, the search-tree has epth at most 2k in which the root has n chilren, an other inner noes have two chilren for each. Thus it has at most 1 + n 2k 1 2i 2 2k n noes. Since the algorithm takes O(2 k k 2 time for checking whether S is secure for each noe of the search-tree, the total running time is O(2 3k k 2 n. 4. Improve Bouns for Hypercubes 4.1 Defensive Alliances in Hypercubes The efensive alliance partition number of G, ψ a (G, is efine to be the maximum number of sets in a partition of V(G such that each set is a efensive alliance [14]. Eroh an Gera [8] showe that ψ a (Q 2 /2, an aske whether the boun is tight. In this section, we first answer the question affirmatively. We then generalize the result to show similar results for Hamming graphs. The first result is an almost irect consequence of the following beautiful proposition. Proposition 4.1 (Chung, Fürei, Graham, an Seymour [2]. If G is a subgraph of a hypercube with average egree s, then V(G 2 s. Theorem 4.2. For any, a(q = 2 /2. Proof. Eroh an Gera [8] showe that a(q 2 /2. Thus we only show the lower boun. Let S be a efensive alliance of Q an v S. Since N[v] S N[v]\S an = N[v] S + N[v]\S 1, it follows that eg Q [S ](v = N[v] S 1 ( 1/2 = /2. Hence the average egree of Q [S ]is at least /2. Now, by Proposition 4.1, S 2 /2. The theorem above implies that ψ a (Q 2 /2 /2 = 2 /2. Combining this fact with the lower boun of Eroh an Gera [8], we have the following answer. Corollary 4.3. For any, ψ a (Q = 2 /2. Now we generalize the result. For Hamming graphs, the following fact is known. Proposition 4.4 (Squier, Torrence, an Vogt [17]. If G is a subgraph of K k with n vertices an m eges, then 2m (k 1n log k n. The proposition above can be seen as a generalization of Proposition 4.1. Corollary 4.5. If G is a subgraph of K k with average egree s, then V(G k s/(k 1. Proof. Assume that G has n vertices an m eges. Then s = 2m/n. By Proposition 4.4, 2m (k 1n log k n. Diviing each sie by (k 1n, we obtain s/(k 1 log k n. This implies k s/(k 1 n.

5 488 Theorem 4.6. For any an k, k /2 a(k k k /2. Proof. First observe that K k is (k 1-regular. To show the upper boun, take a subgraph isomorphic to K /2 k. Each vertex in the subgraph has (k 1 /2 neighbors in the subgraph, an (k 1 /2 neighbors outsie of the subgraph. Thus the vertex set of the subgraph, which is of size k /2, is a efensive alliance. Let S be a efensive alliance of K k.bythesameway as the one in the proof of Theorem 4.2, we can show that the egree of v S is at least (k 1/2. By Corollary 4.5, S k (k 1/2 /(k 1 k /2. Corollary 4.7. For any an k, k /2 ψ a (K k k /2. Proof. The upper boun ψ a (K k k /2 immeiately follows from the lower boun a(k k k /2 in Theorem 4.6, because ψ a (G V(G /a(g for any graph G. The lower boun ψ a (K k k /2 follows from the construction of size k /2 efensive alliances in the proof of Theorem 4.6. The vertices with fixe first /2 coorinates inuce K /2 k. Thus we can partition K k into k /2 isomorphic copies of K /2 k. 4.2 Secure Sets of Hypercubes Petrie [16] showe that 2 /2 s(q 2 1, an aske whether the bouns can be improve. In this section we present an improve lower boun on s(q which is roughly We only use the simple fact that S is not secure if S < N[S ] \ S. Harper [11] showe that the breath-first search orering gives sets S that minimize N[S ] \ S. Using this result, we show that S < N[S ] \ S for any S V(Q with 1 S ( ( 2/3 i. The following proposition follows from Program an Theorem 1 in Harper s paper [11]. Proposition 4.8 (Harper [11]. For any positive integer k 2, there exist a set S V(Q, a vertex u 0 V(Q, an an integer r, such that S = k, N[S ] \ S = min T V(Q, T =k N[T] \ T, an {v ist(u 0,v r} S {v ist(u 0,v r + 1}. Theorem 4.9. For 2, it hols that s(q > ( 2/3 ( i. Proof. First, we show a property of a partial sum over binomial coefficients. Claim For 2, ( r ( i < r+1 for r ( 2/3. We prove the claim by inuction on r. The case of r = 0 clearly hols. Assume ( r 1 ( i < r for some r with 1 r ( 2/3. The assumption r ( 2/3 implies r + 1 2r 1. Therefore, it hols that ( r 1 i < 1 2r 1 = r r + 1 r + 1 1, ( r which implies that r 1 ( < i IEICE TRANS. INF. & SYST., VOL.E98 D, NO.3 MARCH 2015 ( ( ( r r r = r + 1 (. r Thus we have ( r ( i < r+1. Next, by using Proposition 4.8 an Claim 4.10, we can show the following. Claim For any subset S V(Q, if S, then N[S ] \ S > S. ( 2/3 ( i To prove the claim, let k ( ( 2/3 i.lets, u0, an r be the set, the vertex, an the integer in Proposition 4.8, respectively. Recall that {v ist(u 0,v = i} = ( i. Obviously, r ( 2/3. Hence, by Claim 4.10, we have ( r ( i < ( r+1.ifk = r i, then S = {v ist(u0,v r} an N[S ] \ S = {v ist(u 0,v = r + 1}. Thus, the claim hols in this case. In the following, we will focus on the case where k > ( r i. In this case, it hols that r ( 2/3 1 ( 5/3. Let S i S, ist(u 0,v = i}. Clearly, S = S r+1 + r = {v v ( ( < S r+1 +. (1 i r + 1 Let i = (N[S ] \ S {v ist(u 0,v = i}. Then N[S ] \ S = r+1 + r+2. Since S r is exactly the set {v ist(u 0,v = r}, it follows r+1 = ( r+1 Sr+1. Therefore, by (1, S < r S r+1. Now it suffices to show that 2 S r+1 r+2. For any v S r+1, N[v] r+2 = r 1. On the other han, for any v r+2, N[v] S r+1 r + 2. Thus, if F is the set of eges between S r+1 an r+2, then ( r 1 S r+1 = F (r + 2 r+2, which implies (( r 1/(r + 2 S r+1 r+2. Since r ( 5/3 means 2 ( r 1/(r+2, we have 2 S r+1 r+2. Claim 4.11 implies that there is no secure set of size at most ( ( 2/3 i. Therefore, the theorem follows. To see that the lower boun above is a significant improvement, we now show that ( ( 2/3 i > for large enough. (An explicit lower boun will be given in the proof. To this en, we nee the following lower boun of binomial coefficients. Proposition 4.12 (Mitzenmacher an Upfal [15, Corollary 9.3]. For 0 q 1/2, ( 2 H(q q + 1, where H(q is the binary entropy function H(q = q lg q (1 qlg(1 q. Corollary s(q > for large enough.

6 AMANO et al.: SECURE SETS AND DEFENSIVE ALLIANCES 489 Proof. We prove the statement for 652. By Theorem 4.9, it suffices to show that ( ( 2/3 > for 652. Observe that ( 5/3 > 0.33 since > 500, an hence ( ( ( > ( 2/3 ( 5/ , where the last inequality follows from Proposition 4.12 an the fact that H(0.33 > Thus ( ( 2/3 + 1 > It ( is easy to verify that 1.01 > + 1for 652. Hence ( 2/3 > In a similar way, one can show that for any ɛ>0, there is an integer 0 such that for 0, s(q > 2 (H(1/3 ɛ. Note that we nee ifferent argument to show a significantly better lower boun since H(1/3 = <0.92. [15] M. Mitzenmacher an E. Upfal, Probability an computing: Ranomize algorithms an probabilistic analysis, Cambrige University Press, [16] C. Petrie, (F, I-security in graphs, Discrete Appl. Math., vol.162, pp , [17] R. Squier, B. Torrence, an A. Vogt, The number of eges in a subgraph of a Hamming graph, Appl. Math. Lett., vol.14, pp , Kazuyuki Amano receive his M.E. an D.E. egrees in information sciences from Tohoku University in 1993 an 1996, respectively. Presently, he is a professor in Department of Computer Science, Gunma University, Kiryu, Japan. His research interests inclue the theory of computation an combinatorics. Acknowlegments This research was partially supporte by JSPS KAKENHI Grant Numbers , , , an by MEXT KAKENHI Grant Numbers , References [1] R.C. Brigham, R.D. Dutton, an S.T. Heetniemi, Security in graphs, Discrete Appl. Math., vol.155, pp , [2] F.R.K. Chung, Z. Fürei, R.L. Graham, an P.D. Seymour, On inuce subgraphs of the cube, J. Comb. Theory, Ser. A, vol.49, pp , [3] R.D. Dutton, The evolution of a har graph theory problem secure sets, A course material of COT 6410: Computational Complexity, University of Central Floria, Available at [4] R.D. Dutton, On a graph s security number, Discrete Math., vol.309, pp , [5] R.D. Dutton, R. Lee, an R.C. Brigham, Bouns on a graph s security number, Discrete Appl. Math., vol.156, pp , [6] R.I. Enciso an R.D. Dutton, Algorithms for efensive alliance problems, Available at renciso/da.pf [7] R.I. Enciso an R.D. Dutton, Parameterize complexity of secure sets, Congr. Numer., vol.189, pp , [8] L. Eroh an R. Gera, Alliance partition number in graphs, Ars Comb., vol.103, pp , [9] H. Fernau an D. Raible, Alliances in graphs: A complexitytheoretic stuy, SOFSEM (2, pp.61 70, [10] G.H. Fricke, L.M. Lawson, T.W. Haynes, S.M. Heetniemi, an S.T. Heetniemi, A note on efensive alliances in graphs, Bull. Inst. Combin. Appl., vol.38, pp.37 41, [11] L.H. Harper, Optimal numberings an isoperimetric problems on graphs, J. Combin. Theory, vol.1, pp , [12] G. Isaak, P. Johnson, an C. Petrie, Integer an fractional security in graphs, Discrete Appl. Math., vol.160, pp , [13] L.H. Jamieson, S.T. Heetniemi, an A.A. McRae, The algorithmic complexity of alliances in graphs, J. Combin. Math. Combin. Comput., vol.68, pp , [14] P. Kristiansen, S.M. Heetniemi, an S.T. Heetniemi, Alliances in graphs, J. Combin. Math. Combin. Comput., vol.48, pp , Kyaw May Oo receive her B.Sc. in Mathematics from Yangon University; M.I.Sc. from Institute of Computer Science an Technology (ICST; an Ph.D. in Information Science from University of Computer Stuies, Yangon (UCSY, Myanmar in 1994, 1996, an 2007 respectively. Currently, she is an associate professor in the Department of Computational Mathematics, UCSY, Myanmar. Her research interests inclue graph theory, an ata mining. Yota Otachi is an assistant professor of Japan Avance Institute of Science an Technology. He receive B.E., M.E., an Ph.D. egrees from Gunma University in 2005, 2007, an 2010, respectively. His research interests inclue graph algorithms, graph theory, an complexity theory. Ryuhei Uehara receive B.E., M.E., an Ph.D. egrees from the University of Electro- Communications, Japan, in 1989, 1991, an 1998, respectively. He was a researcher in CANON Inc. uring In 1993, he joine Tokyo Woman s Christian University as an assistant professor. He was a lecturer uring , an an associate professor uring at Komazawa University. He move to Japan Avance Institute of Science an Technology (JAIST in 2004 as an associate professor, an since 2011, he has been a professor in School of Information Science. His research interests inclue computational complexity, algorithms, an ata structures, especially, ranomize algorithms, approximation algorithms, graph algorithms, an algorithmic graph theory. He is a member of EATCS, ACM, an IEEE.