The minimum identifying code graphs

Discrete Applied Mathematics 160 (2012) 1385–1389 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The minimum identifying code graphs André Raspaud a , Li-Da Tong b,∗ a b LaBRI U.M.R. 5800, Université Bordeaux I, 351 Cours de la Libération, F33405 Talence Cedex, France Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan article info Article history: Received 10 July 2010 Received in revised form 5 January 2012 Accepted 15 January 2012 Available online 17 February 2012 Keywords: Identifying code Graph Power set abstract Let G be a graph and B(u) be the set of u with all of its neighbors in G. A set S of vertices is called an identifying code of G if, for every pair of distinct vertices u and v , both B(u) ∩ S and B(v) ∩ S are nonempty and distinct. A minimum identifying code of a graph G is an identifying code of G with minimum cardinality and M (G) is the cardinality of a minimum identifying code for G. A minimum identifying code graph G of order n is a graph with M (G) = ⌈log2 (n + 1)⌉ having the minimum number of edges. Moncel (2006) [5] constructed minimum identifying code graphs of order 2m − 1 for m ≥ 2 and left the same problem but for arbitrary order open. In this paper, we aimed at the construction of connected minimum identifying code graphs in order to solve this problem for integer order greater than or equal to 4. Furthermore, we discussed some related properties. © 2012 Elsevier B.V. All rights reserved. 1. Introduction The identifying codes were first introduced by Karpovsky et al. in [3]. Identifying codes of graphs are constructed for the diagnosis of faults in multiprocessor systems [2–4]. Charon et al. [1] proved that determining an identifying code with minimum cardinality in a graph is NP-hard. Let G be a graph, u be a vertex of G, and B(u) (or BG (u)) be the set of u with all its neighbors in G. A set S of vertices is called an identifying code of G if, for every pair of distinct vertices u and v, B(u) ∩ S and B(v) ∩ S are nonempty and distinct. A minimum identifying code of a graph G is an identifying code of G with minimum cardinality and M (G) is the cardinality of a minimum identifying code in G. A minimum identifying code graph G of order n is a graph with M (G) = ⌈log2 (n + 1)⌉ having the minimum number of edges. Finding the number of edges on a minimum identifying code graph is an important problem in reducing the cost of constructing a minimum identifying code graph or a network. Moncel [5] constructed minimum identifying code graphs of order 2m − 1 for each integer m ≥ 2 and addressed the question of finding minimum identifying code graphs of order n for arbitrary positive integer n. In this paper, we construct minimum identifying code graphs of order n for each positive integer n ≥ 4 and investigate related properties. Hence, this question is completely solved. 2. Minimum identifying code graphs In this section, we construct minimum identifying code graphs. The first proposition is to investigate the lower bound of M (G) for a graph G. It was proved by Karpovsky, Chakrabarty, and Levitin. Proposition 1 ([3]). If G is a graph of order n, then M (G) ≥ ⌈log2 (n + 1)⌉. In [3], it is shown that there exists a graph G of order n with M (G) = ⌈log2 (n + 1)⌉ for each positive integer n ≥ 4. The following gives a simple procedure for constructing such graphs. Let S be an m-set and P (S ) be the power set of S. Take a ∗ Corresponding author. E-mail address: ldtong@math.nsysu.edu.tw (L.-D. Tong). 0166-218X/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2012.01.015 1386 A. Raspaud, L.-D. Tong / Discrete Applied Mathematics 160 (2012) 1385–1389 subset V of P (S ) − {∅} satisfying 2m−1 ≤ |V | < 2m and {x} ∈ V for any x ∈ S. Define a graph H (V ) as the graph with vertex set V and edge set {{x}A : x ∈ A and |A| ≥ 2}. Let T be the set {{x} : x ∈ S }. If x ∈ C − D for C , D ∈ V , then B(C ) ∩ T and B(D) ∩ T are distinct in H (V ). This implies that T is an identifying code of H (V ) and |T | = m. Thus there exists a minimum identifying code graph for any order ≥4. Next, we give a lower bound for the number of edges of a minimum identifying code graph. Let A1 , A2 , . . . , A2m −1 all be nonempty subsets of an m-set S with ai = |Ai | and a1 ≤ a2 ≤ · · · ≤ a2m −1 . n Proposition 2. If G is a minimum identifying code graph of order n ≥ 4 and m = ⌈log2 (n + 1)⌉, then |E (G)| ≥ ⌈  ( nj=n−m+1 aj /2) − m/2⌉. i=1 ai − Proof. Suppose that G = (V , E ) is a minimum identifying code graph of order n and S is a minimum  identifying code for G.Then there are no edges between two distinct v∈V −S |B(v) ∩ S | +  vertices in V − S. This implies that |E (G)| = 1/2( u∈S (|B(u) ∩ S | − 1)) = | B (v) ∩ S | − | B ( u ) ∩ S |/ 2 − m / 2. Since, for v ∈ V , all B (v) ∩ S’s are distinct and v∈ u∈S Vn n nonempty, |E (G)| ≥ ⌈ i=1 ai − ( j=n−m+1 aj /2) − m/2⌉.  n Corollary 3. If G is a graph of order n ≥ 4 and M (G) = m, then |E (G)| ≥ ⌈ i=1 ai − ( n j=n−m+1 aj /2) − m/2⌉. Let m, t, and k be three positive integers. An (m, t , k)-graph is a graph of order m with t vertices of degree k and m − t vertices of degree k − 1 such that for any pair of distinct vertices u and v, B(u) and B(v) are distinct. Theorem 4. Let m, t , k be positive integers with m ≥ 3, t ≤ m and 2 ≤ k ≤ m − 1. Then there exists an (m, t , k)-graph if and only if tk + (m − t )(k − 1) is even and (m, t , k) ̸∈ {(m, t , m − 1) : t ≥ 2} ∪ {(m, t , 2) : 2t + 2 ≤ m − t }. Proof. Suppose that there exists an (m, t , k)-graph. Since the degree sum of a graph must be even, tk+(m−t )(k−1) is even. If a graph of order n has at least two vertices with degree n − 1, then it has no identifying codes; that is, (m, t , k) ̸= (m, t , m − 1) for t ≥ 2. If 2t + 2 ≤ m − t, then an (m, t , 2)-graph would have at least one independent edge; that is, it has no identifying code, a contradiction. Conversely, we construct (m, t , k)-graphs satisfying the conditions of this theorem. For k = 2 and 2t + 2 > m − t, we can find a graph G as a union of some disjoint paths of length at least 2. Trivially, G is an (m, t , 2)-graph. Since a cycle of length at least 4 has an identifying code, there exists an (m, m, 2)-graph for m ≥ 4. If there is an (m, 1, m − 1)-graph, then m − 1 is even. Let G be a graph obtained from Km by deleting a maximum matching. It is easy to see that V (G) is an identifying code of G. Now, there only remain the cases m ≥ 5 and 3 ≤ k ≤ m − 2. Case 1. k is even. Let k = 2r with r ≥ 2 and G = G(m, r ) = ({0, 1, 2, . . . , m−1}, {ij : i−j ≡ a (mod m) for some 1 ≤ a ≤ r }). It is easy to check that B(u) ̸= B(v) for any vertices u ̸= v . And we have the property that if there exist two distinct vertices u and v in G satisfying that there is only one vertex x ∈ B(u) − B(v), then either |u − v| = 1 or (x = u and m = k + 2). Then for the case of |u − v| = 1, without loss of generality, let v ≡ u + 1 (mod m). Then, x ≡ u − r (mod m). By the symmetry of the graph, there exists y ∈ B(v) − B(u) with y ≡ u + 1 + r (mod m). Let ei = (2i)(2i + 1) for i = 0, 1, 2, . . . , ⌊m/2⌋ − 1 and Gi = G − {e0 , . . . , ei−1 } for i = 1, 2, . . . , ⌊m/2⌋ . We can observe that there exists at most one vertex y with |x − y| = 1 in BG (x) − BGi (x) for each vertex x. Since r ≥ 2, we have that Gi has 2i vertices of degree k − 1, m − 2i vertices of degree k, and an identifying code V (G). For the case of x = u and m = k + 2, Gi is still a graph having an identifying code, and having 2i vertices of degree k − 1, and m − 2i vertices of degree k. Case 2. k is odd. If there is an (m, t , k)-graph, then t is even. The following numbers are taken modulo m. Let k − 1 = 2r ≤ m − 3, p = ⌊m/2⌋, and G = G(m, r ), defined in Case 1. Define that ei = i(i + p) and Gi = G + {e1 , e2 , . . . , ei } for i = 0, 1, 2, . . . , p − 1. Then Gi has 2i vertices of degree k and m − 2i vertices of degree k − 1. We claim that Gi has an identifying code V (G) for all i. Indeed, take two vertices u and v in G. Without loss of generality, v = u + d (mod m) for some positive integer d ≤ p. If 1 ≤ d ≤ r, then u − r ∈ BGi (u) − BGi (v) is nonempty. If d ≥ r + 1, then u ∈ BGi (u) − BGi (v). Thus Gi has an identifying code V (G) for all i.  Let S = {1, 2, . . . , m}, and < be a linear ordering of P (S ) − {∅} defined by A < B if and only if either |A| < |B| or, for |A| = |B|, the smallest element of A − B is smaller than that of B − A. Then we have that P (S ) = {A1 , A2 , . . . , A2m −1 } with Ai < Aj for i < j. Define ai = |Ai | for i = 1, 2, . . . , 2m − 1 and let n be a positive integer with 2m−1 ≤ n ≤ 2m − 1 and m ≥ 3. Next, we construct a minimum identifying code graph Hn of order n. If n ≤ 10 (respectively, n ≥ 11) and m = ⌈log2 (n + 1)⌉, then an ≤ 2 (respectively, an ≥ 3). For n ≥ 11 n m−1 and ≤ n ≤ 2m − 1, there exists a graph Gn with V (Gn ) = S i=n−m+1 (ai − 1) being even, by Theorem 4 and 2 such that |B(i) ∩ S | = an−i+1 for i = 1, 2, . . . , m and B(1), B(2), . . . , B(m) are distinct. Then we can choose Bi from P (S ) − {∅} with |Bi | = ai for i = 1, 2, . . . , n − m such that B1 , . . . , Bn−m , B(1), B(2), . . . , B(m) are distinct. Define Hn = (S ∪ {B1 , . . . , Bn−m }, E (Gn ) ∪ {xBi : x ∈ S and x ∈ Bi }). It is trivial that BHn (Bi ) = Bi for all i. Then we have that S is an identifying code of Hn . n−1 n For n ≥ 11, i=n−m+1 (ai − 1) being odd and an = m, we have n = 2m − 1 and an−m = m − 1. Then i=n−m (ai − 1) is even. By Theorem 4 and 2m−1 ≤ n ≤ 2m − 1, there exists a graph Gn with V (Gn ) = S such that |B(i)∩ S | = an−i for 1 ≤ i ≤ m and B(1), B(2), . . . , B(m) are distinct. Take Bi from P (S ) − {∅} with |Bi | = ai for i = 1, 2, . . . , n − m − 1 and Bn−m = S such A. Raspaud, L.-D. Tong / Discrete Applied Mathematics 160 (2012) 1385–1389 1387 that B1 , . . . , Bn−m , B(1), B(2), . . . , B(m) are distinct. Let Hn = (S ∪ {A1 , . . . , An−m−1 , An }, E (Gn ) ∪ {xAi : x ∈ Ai }). Then it is easy to check that nS is an identifying code of Hn . For n ≥ 11, i=n−m+1 (ai − 1) being odd and an < m, there exists a largest j ∈ {n − m + 1, . . . , n} such that aj = an−m+1 . By Theorem 4, there exists a graph Gn with V (Gn ) = S such that |B(i)| = ai for i ∈ {1, 2, . . . , m} − {j}, |B(j)| = aj + 1, and B(1), B(2), . . . , B(m) are distinct. Take Bi with |Bi | = ai for i = 1, 2, . . . , n − m such that B1 , . . . , Bn−m , B(1), B(2), . . . , B(m) are distinct. Let Hn = (S ∪ {B1 , . . . , Bn−m }, E (Gn ) ∪ {xBi : x ∈ Bi }). Then we have that S is an identifying code of Hn . Theorem 5. For each positive integer n ≥ 11 with 2m−1 ≤ n < 2m , Hn is a minimum identifying code graph of order n. n Proof. Let n be a positive integer with 2m−1 ≤ n < 2m and n ≥ 11. If i=n−m+1 (ai − 1) is even, then the number of edges n n of Hn is i=1 ai − ( j=n−m+1 aj /2) − m/2. By the above discussion of Hn and Proposition 2, Hn is a minimum identifying code graph. n−1 n n n n If i=n−m+1 (ai − 1) is odd and an = m, then ⌈ i=1 ai −( j=n−m+1 aj /2)− m/2⌉ = ( i=1 ai )+ 1 −(( j=n−m+1 aj )+ an − n n n n a /2) − m/2⌉. If i=n−m+1 (ai − 1) is odd and an < m, then ⌈ i=1 ai − a −( n n n i=1 i n j=n−m+1 j n ( j=n−m+1 aj /2)− m/2⌉ = ( i=1 ai )−(( j=n−m+1 aj )+ 1)/2 − m/2; that is, |E (Hn )| = ⌈ i=1 ai −( j=n−m+1 aj /2)− m/2⌉. n Since M (Hn ) = m and Proposition 2, Hn is also a minimum identifying code graph for i=n−m+1 (ai − 1) being odd. Thus we 1)/2 − m/2; that is, |E (Hn )| = ⌈ have the theorem.  By Proposition 2 and Theorem 5, we have the following theorem. n Theorem 6. If G is a minimum identifying code graph of order n ≥ 11 and m = ⌈log2 (n + 1)⌉, then |E (G)| = ⌈  ( nj=n−m+1 aj /2) − m/2⌉. i=1 ai − Remarks. For minimum identifying code graphs of order 4 ≤ n ≤ 10, we discuss the numbers of their edges in the remarks. (a) For n = 4, let H4 = ({1, 2, 3, 4}, {23, 34}). Then {1, 2, 4} is an identifying code for H4 . By Proposition 2, it is a minimum identifying code graph. (b) By Proposition 2, we have a lower bound 3 for the number of edges of a minimum identifying code graph of order 5. If there is a minimum identifying code graph G of order 5 and size 3, and S is a minimum identifying code, then the induced subgraph of S in G is either a path of length 2 or without edge. This implies that either there exists an isolated vertex in V (G) − S, or there exist four vertices of degree 1 in G. This contradicts that S is an identifying code in G. Let H5 = ({1, 2, 3, 4, 5}, {12, 23, 34, 45}) and S = {2, 3, 4}. Then S is a minimum identifying code for H5 . So the number of edges of a minimum identifying code graph with five vertices is 4. (c) For n = 6, let H6 = ({1, 2, 3, x, y, z }, {12, 23, 1x, 2y, 3z }). Then {1, 2, 3} is a minimum identifying code of H6 . By Proposition 2, it is a minimum identifying code graph. (d) For n = 7, let H7 = ({1, 2, 3, x, y, z , w}, {12, 23, 1x, 2y, 3z , 1w, 3w}). Then {1, 2, 3} is a minimum identifying code of H7 . By Proposition 2, it is a minimum identifying code graph. (e) For n = 8, if there is a minimum identifying code graph G of order 8 and size 6, and T is a minimum identifying code, then the induced subgraph G′ of T in G contains at most two edges. If G′ has some edges, then this contradicts that T is an identifying code. Thus the edge set of G′ is empty, and if BG (v) ∩ T = {u} then v ∈ T . So every vertex in V (G) − T is adjacent to at least two distinct vertices of T . Then |E (G)| ≥ 8, a contradiction. Therefore if G is a minimum identifying code graph G of order 8, then |E (G)| ≥ 7. Let H8 = ({1, 2, 3, 4, x, y, z , w}, {12, 23, 34, 1x, 2y, 3z , 4w}). Then {1, 2, 3, 4} is a minimum identifying code of H8 . By the above, it is a minimum identifying code graph of order 8. (f) By Proposition 2, 8 is a lower bound for the number of edges of a minimum identifying code graph of order 9. If there is a minimum identifying code graph G of order 9 and size 8, and S is a minimum identifying code, then the induced subgraph of S in G is either a path of length 2 or without edge. This implies that the number of edges in G is at least 9, a contradiction. Let H9 = ({1, 2, 3, 4, x, y, z , w, v}, {12, 23, 34, 1x, 2y, 3z , 4w, 1v, 4v}). Then {1, 2, 3, 4} is a minimum identifying code of H9 . Thus H9 is a minimum identifying code graph of order 9 and size 9. (g) By Proposition 2, 10 is a lower bound for the number of edges of a minimum identifying code graph of order 10. If there is a minimum identify code graph G of order 10 and size 10, and S is a minimum identifying code, then the induced subgraph of S in G is either a path of length 2 or without edge. This implies that the number of edges in G is at least 12, a contradiction. Let H10 = ({1, 2, 3, 4, x, y, z , w, v, u}, {12, 23, 34, 1x, 2y, 3z , 4w, 1v, 4v, 1u, 3u}). Then {1, 2, 3, 4} is a minimum identifying code of H10 . Thus H10 is a minimum identifying code graph of order 10 and size 11. Proposition 7. If G is a minimum identifying code graph of order n ≥ 6 and S is a minimum identifying code of G, then S is a minimum dominating set and V (G) − S is a maximum independent set in G. Proof. Let G be a minimum identifying code graph of order n ≥ 6 and S be a minimum identifying code of G. By the definition of a minimum identifying code graph, S is a dominating set and V (G) − S is an independent set in G. Since G has exactly ⌈ ni=1 ai − ( nj=n−m+1 aj /2) − m/2⌉ edges, we have that for each x ∈ S , |B(x) ∩ S | ≥ 2; that is, there exists v ∈ V (G) − S such that v is a leaf of G and v x ∈ E (G). Then S must be a minimum dominating set. Let I be a maximum independent set 1388 A. Raspaud, L.-D. Tong / Discrete Applied Mathematics 160 (2012) 1385–1389 of G. If there exists a vertex x ∈ S ∩ I, then there exists v ∈ V (G) − S such that v is a leaf of G and v x ∈ E (G). Consequently, we have that I − {x} ∪ {v} is a maximum independent set, too. This implies that there exists a maximum independent set without any vertices in S. We conclude that V (G) − S is a maximum independent set of G.  Remark. We can observe that (a) the size of H2m −1 is m(2m−1 − 1) − (m 2 ) + ⌊m/2⌋, and (b) the diameter of Hn is 4 for n ≥ 6. 3. The k-minimum identifying code graphs In Theorem 6, it is shown that for every minimum identifying code graph G and a minimum identifying code T of G, there exists a vertex v of G such that |B(v) ∩ T | = 1. A graph G of order n with M (G) = ⌈log2 (n + 1)⌉ is called a k-minimum identifying code graph if (i) for every minimum identifying code T , |B(v) ∩ T | ≥ k for each vertex v ∈ V (G), and (ii) the number of edges of G is as small as possible. Then every minimum identifying code graph is a 1-minimum identifying code graph. In this section, our concern is how to construct k-minimum identifying code graphs for k ≥ 2. Proposition 8. Suppose that k ≥ 2 is a positive integer. If G is a k-minimum identifying code graph of order n ≥ 6 and T is a minimum identifying code, then V (G) − T is an independent set. Proof. Since G is a k-minimum identifying code graph of order n ≥ 6, if there exist two distinct vertices u, v ∈ V (G) − T with uv ∈ E (G), then for each vertex x of H = G − {uv}, |BH (x) ∩ T | ≥ k and |E (H )| < |E (G)|, a contradiction. Thus, V (G) − T must be an independent set.  Theorem 9. Suppose that G is a k-minimum identifying code graph of order n ≥ 4. Let m = ⌈log2 (n + 1)⌉, T be a minimum identifying  code of G, and  V (G) = {v1 , v2 , . . . , vn } with bi = |B(vi ) ∩ T | for i = 1, 2, . . . , n. If b1 ≤ b2 ≤ · · · ≤ bn , then |E (G)| ≥ ⌈ im=−1n bi + ( nj=n−m+1 (bj − 1)/2)⌉. Proof. By Proposition 8, each pair of distinct vertices in V (G) − T are nonadjacent. This implies nevery vertex m−n that for x ∈ V (G) − T , B(x) − {x} ⊆ T . Therefore, the number of edges of G is greater than or equal to ⌈ i=1 bi + ( j=n−m+1 (bj − 1)/2)⌉.  Theorem 10. Let n and m be two positive integers with m = ⌈log2 (n + 1)⌉. Suppose that k ≥ 2 is a positive integer, t is the smallest number such that at = k, and t + n − 1 ≤ 2m − 1. If G is a k-minimum identifying code graph of order n ≥ 6, then   |E (G)| = ⌈ in=−0m−1 at +i − ( nj=−n1−m at +j /2) − m/2⌉ for k ≤ m − 2. Proof. Suppose that G is a k-minimum identifying code graph of order n ≥ 6 and T is a minimum identifying code of G. By Theorem 9,let V (G) ={v1 , v2 , . . . , vn } with bi = |B(vi ) ∩ T | for i = 1, 2, . . . , n and b1 ≤ b2 ≤ · · · ≤ bn ; we have that |E (G)| ≥ ⌈ im=−1n bi + ( nj=n−m+1 (bj − 1)/2)⌉. By Proposition 8 and the fact that B(vi ) ∩ T , for i = 1, 2, . . . , n, are all distinct, m−n n−m−1 n n−1 at +i − ( j=n−m at +j /2) − m/2⌉. we have that ⌈ i=1 bi + ( j=n−m+1 (bj − 1)/2)⌉ ≥ ⌈ i=0 Since n ≥ 6, we have that n ≥ 2m and k ≤ m − 2. By a discussion similar to that in Theorem 5, we have a k-minimum n−m−1 n−1 identifying code graph of order n with ⌈ i=0 at +i − ( j=n−m at +j /2) − m/2⌉ edges for k ≤ m − 2.  We can observe that the k-minimum identifying code graphs in Theorem 10 are (k − 1)-connected or k-connected. Thus, we focus on k-connected graphs G of order n with M (G) = ⌈log2 (n + 1)⌉. Let T = {0, 1, . . . , m − 1} and Bi = {i, i + 1, . . . , i + k − 1} for i = 0, 1, . . . , m − 1 where the numbers are taken modulo m. Define G(m, k) as a graph with V (G(m, n)) = S ∪ {B1 , B2 , . . . , Bm } and E (G(m, n)) = {xBi : x ∈ Bi }. Thus, G(m, k) is a k-connected bipartite graph of order 2m. Proposition 11. Let n, m ≥ 4, k ≥ 2 be positive integers with m = ⌈log2 (n + 1)⌉ and t be the smallest number with at = k. If n ≤ 2m + m − t, then there exists a k-connected graph of order n with M (G) = m. Proof. Let C1 , C2 , . . . , Cn−2m be distinct subsets of T = {0, 1, . . . , m − 1} satisfying |Ci | ≥ k and Ci ̸∈ V (G(m, n)) for all i. Define Hnk as a graph with V (Hnk ) = V (G(m, k)) ∪ {C1 , C2 , . . . , Cn−2m } and E (Hnk ) = E (G(m, k)) ∪ {xCj : x ∈ Cj }. Since G(m, k) is k-connected, degH k (Cj ) ≥ k, and BH k (Cj ) − {Cj } ⊆ T , Hnk is k-connected. Also BH k (x) ∩ T = {x}, BH k (Bi ) ∩ T = Bi , and n n n n BH k (Cj ) ∩ T = Cj for x ∈ T , 1 ≤ i ≤ m, and 1 ≤ j ≤ n − 2m. Then T is an identifying code of Hnk . Thus we have a k-connected n graph of order n with M (G) = m.  In Section 1, we have a connected graph G of order n with M (G) = ⌈log2 (n + 1)⌉ and minimum size. We prove that there exists a k-connected graph of order n with M (G) = ⌈log2 (n + 1)⌉ in Proposition 11. In the following, we discuss the 2-connected graph G of order n with minimum size and M (G) = ⌈log2 (n + 1)⌉. A. Raspaud, L.-D. Tong / Discrete Applied Mathematics 160 (2012) 1385–1389 1389 Proposition 12. Let n be an integer  with n ≥ 9 and m = ⌈log2 (n + 1)⌉. If G is a 2-connected graph of order n with minimum n n size and M (G) = m, then |E (G)| ≥ ⌈ i=1 ai − j=n−m+1 aj /2⌉. Proof. For each integer n ≥ 9, n > 2m. Assume that G is a 2-connected graph of order n with minimum size and M (G) = ⌈log2 (n+1)⌉, and let T be a minimum identifying code of G. Let V1 = {u ∈ V (G) : |B(u)∩T | = 1 and u ∈ T } and V2 = {u ∈ V (G) : |B(u) ∩ T | = 1 and u ̸∈ T }. Without loss of generality, let V (G) = 1 , v2 , . . . , vn } and Tn = {vn−m+1 , . . . , vn }. {v n Let bi = |B(vi ) ∩ T | for i = 1, 2, . . . , n. Since G is 2-connected, |E (G)| ≥ ⌈ i=1 bi + |V2 |/2 − j=n−m+1 bj − m/2⌉. In n 2 − m/2) is greater than or equal to j=n−m+1 aj /  (m − |V1 | − |V2 |)/2 + |V1 |/2 + |V2 |/2 = m/2. This implies that E (G) ≥ ⌈ ni=1 ai − nj=n−m+1 aj /2⌉.  n addition, ( i=1 bi + |V2 |/2 − n j=n−m+1 n bj /2 − m/2) − ( i=1 ai − Theorem 13. Let n be an integer with n ≥ 6 and m = ⌈log2 (n+ 1)⌉. If an−m+1 ≥ 3 and G is a 2-connected graph of order n n n with minimum size and M (G) = m, then |E (G)| ≤ ⌈ i=1 ai − j=n−m+1 aj /2 − m/2⌉ + ⌈m/2⌉. Proof. We consider the graph Hn defined in Section 2. Since an−m+1 ≥ 3, the induced subgraph of {1, 2, . . . , m} ∪ {{x, y} : x ̸= y and x, y ∈ {1, 2, . . . , m}} in Hn is 2-connected. The vertices, except the singletons, have two distinct neighbors in {1, 2, . . . , m}. Thus, if A is the set of all singletons in {1, 2, . . . , m}, then Hn − A is 2-connected. Let A = {u1 , u2 , . . . , um }. If m is even, then set B = {u2i−1 u2i : i = 1, 2, . . . , m/2}; otherwise, set B = {u2i−1 u2i : i = 1, 2, . . . , (m − 1)/2} ∪ {u1 um }. Define Hn+ as with V (Hn+ ) = V (Hn ) and E (Hn+ ) = E (Hn ) ∪ B. It is easy to check that Hn+ is 2-connected and an graph  + |E (Hn )| = ⌈ i=1 ai − nj=n−m+1 aj /2 − m/2⌉ + ⌈m/2⌉. Then we have the result that if an−m+1 ≥ 3 and G is a 2-connected n graph of order n with minimum size and M (G) = m, then |E (G)| ≤ ⌈ i=1 ai − n j=n−m+1 aj /2 − m/2⌉ + ⌈m/2⌉.  Corollary 14. Let n be an integer with n ≥ 6 and m = ⌈log be even. If an−m+1 ≥ 3 and G is a 2-connected graph of 2 (n + 1)⌉  n n order n with minimum size and M (G) = m, then |E (G)| = ⌈ i=1 ai − j=n−m+1 aj /2⌉. n n n Proof. Since m is even, ⌈ i=1 ai − j=n−m+1 aj /2 − m/2⌉ + ⌈m/2⌉ = ⌈ is a 2-connected graph of order n with minimum size and M (G) = m.  i=1 ai − n j=n−m+1 aj /2⌉. By Proposition 12, Hn+ In Corollary 14, we give a class of 2-connected graphs of order n with minimum size and M (G) = ⌈log2 (n + 1)⌉. Further study will be focused on the minimum size of a k-connected graph of order n with M (G) = ⌈log2 (n + 1)⌉ for k ≥ 2. Acknowledgments The authors thank the referee for many constructive suggestions. This research was partially supported by the National Science Council under grants NSC 99-2918-I-110-001 and 99-2923-M-002-007-MY3, National Center of Theoretical Sciences. References [1] I. Charon, O. Hudry, A. Lobstein, Minimizing the size of an identifying or locating–dominating code in a graph is NP-hard, Theoret. Comput. Sci. 290 (3) (2003) 2109–2120. [2] S. Gravier, J. Moncel, Construction of codes identifying sets of vertices, Electron. J. Combin. 12 (2005) R13. [3] M.G. Karpovsky, K. Chakrabarty, L.B. Levitin, On a new class of codes for identifying vertices in graphs, IEEE Trans. Inform. Theory 44 (2) (1998) 599–611. [4] T. Laihonen, S. Ranto, Codes identifying sets of vertices, Lecture Notes in Comput. Sci. 2227 (2001) 82–91. [5] J. Moncel, On graphs on n vertices having an identifying code of cardinality ⌈log2 (n + 1)⌉, Discrete Appl. Math. 154 (14) (2006) 2032–2039.