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Theoretical Computer Science www.elsevier.com/locate/tcs

The hub number of co-comparability graphs ✩ Jia-Jie Liu a,∗ , Cindy Tzu-Hsin Wang b , Yue-Li Wang b , William Chung-Kung Yen a a b

Department of Information Management, Shih Hsin University, Taipei, Taiwan Department of Information Management, National Taiwan University of Science and Technology, Taipei, Taiwan

a r t i c l e

i n f o

Article history: Received 31 March 2014 Received in revised form 29 October 2014 Accepted 19 December 2014 Available online 29 December 2014 Communicated by G.F. Italiano Keywords: Dominating set Hub number Connected hub number Co-comparability graph

a b s t r a c t A set H ⊆ V is a hub set of a graph G = ( V , E ) if, for every pair of vertices u , v ∈ V \ H, either u is adjacent to v or there exists a path from u to v such that all intermediate vertices are in H. The hub number of G, denoted by h(G ), is the minimum size of a hub set in G. The connected hub number of G, denoted by hc (G ), is the minimum size of a connected hub set in G. In this paper, we prove that h(G ) = hc (G ) for co-comparability graphs G and characterize the case for which γc (G ) = hc (G ) in this class of graphs, where γc (G ) denotes the connected domination number of G. We also show that h(G ) can be computed in O (| V |) time for trapezoid graphs and in O (| V |3 ) time for co-comparability graphs. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Let G = ( V , E ) be a simple connected graph with vertex set V (G ) and edge set E (G ). Sets V (G ) and E (G ) are simply written as V and E, respectively, when it is clear from context. Suppose that H ⊆ V and u and v are two vertices in V \ H . A path from u to v is called an H -path, denoted by hp H (u , v ), if all intermediate vertices are from H . Note that any edge uv and single vertex u are the degenerated cases of H -paths and are called trivial H -paths. In [15], Walsh deﬁned hub sets as follows: Deﬁnition 1.1. A set H ⊆ V is a hub set of G if, for every pair of vertices u , v ∈ V \ H , there exists an hp H (u , v ). The hub number of G, denoted by h(G ), is the minimum size of a hub set in G. A hub set H is connected if the subgraph induced by H is connected in G. The connected hub number of G, denoted by hc (G ), is the minimum size of a connected hub set in G. Walsh proved that determining whether a graph G has a (connected) hub set of size k is NP-complete [15]. The problem of ﬁnding the hub number of a graph can be applied to networks, e.g., rapid-transit systems (RTS), so that every vertex in the network at which no hub is allocated is adjacent to a vertex with a hub [15]. This system can be regarded as that the distance between any two vertices is at most two if the time spent in RTS can be neglected. A dominating set in a graph G is a subset D of V such that every vertex in V \ D has at least one adjacent vertex in D. A connected dominating set is a dominating set whose induced subgraph is connected in G. The minimum size of a

✩ This work was supported in part by the National Science Council of the Republic of China under Contracts NSC 102-2221-E-128-002-, NSC 101-2221-E-011-038-MY3, and NSC 100-2221-E-128-004-. Correspondence to: J.-J. Liu, Department of Information Management, Shih Hsin University, 1 Lane 17, Sec. 1, Mu-Cha Rd., Taipei, Taiwan 10607, Republic of China. E-mail address: [email protected] (J.-J. Liu).

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http://dx.doi.org/10.1016/j.tcs.2014.12.011 0304-3975/© 2014 Elsevier B.V. All rights reserved.

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Fig. 1. S = A h .

dominating (respectively, connected dominating) set in G, denoted by γ (G ) (respectively, γc (G )), is the domination number (respectively, connected domination number) of G. In [15], Walsh proved that γ (G ) h(G ) + 1 and hc (G ) γc (G ). Later, Grauman et al. showed that γc (G ) h(G ) + 1 [6]. This yields the consecutive inequalities γ (G ) − 1 h(G ) hc (G ) γc (G ) h(G ) + 1 [6]. By the inequality above, it is easy to see that γc (G ) hc (G ) + 1. In [8], Johnson, Slater, and Walsh characterized the class of graphs G satisfying the equality γc (G ) = hc (G ) + 1. The hub number is a newly introduced graph invariant and it is known that its determination is NP-hard. Hence it makes sense to determine this invariant exactly for some interesting and non-trivial graph classes, e.g., grid graphs [7] and ´ Sierpinski-like graphs [13]. In this paper, we are concerned with the hub number of co-comparability graphs and trapezoid graphs. The rest of this paper is organized as follows. Section 2 gives some suﬃcient conditions for h(G ) = hc (G ) in general graphs. In Section 3, we prove that h(G ) = hc (G ) in co-comparability graphs G. Moreover, we also characterize the cases for which γc (G ) = hc (G ). Finally, concluding remarks are given in Section 4. 2. Some suﬃcient conditions for h( G ) = hc ( G ) in general graphs Assume that H is a minimum hub set of graph G and S = V \ H . Hereafter, maximal connected components are simply written as components and the number of components reduced by H is denoted by c ( H ). Let H 1 , H 2 , . . . , H c ( H ) be the maximal connected components induced by H . We also use H i to denote V ( H i ) if no confusion is possible. For a vertex v ∈ V , let N S ( v ) = {u |u ∈ S , uv ∈ E } and N S [ v ] = N S ( v ) ∪ { v } if v ∈ S. Furthermore, let N S ( H i ) = v ∈ V ( H i ) N S ( v ) for 1 i c ( H ). If S = V , then N S ( v ) and N S [ v ] are simply written as N ( v ) and N [ v ], respectively. Since G is connected, | N S ( H i )| > 0 when H is not empty. Let A s = { v | N S [ v ] = S }, A h = { v | v ∈ N S ( H i ) for 1 i c ( H )}, and A o = S \ ( A h ∪ A s ). For a set H , H + v denotes H ∪ { v } and H − v denotes H \ { v }. Lemma 2.1. Assume that H is a minimum hub set of a connected graph G. If A h = V \ H , then h(G ) = hc (G ). Proof. If c ( H ) = 1, then h(G ) = hc (G ) follows directly. Thus we assume that c ( H ) > 1 and H 1 , H 2 , . . . , H c ( H ) are the components of the graph induced by H . Let v 1 ∈ A h and v a ∈ V ( H 1 ) such that the graph induced by H 1 + v 1 − v a is still connected, where A h = V \ H . Let H = H + v 1 − v a and S = A h + v a − v 1 . Since the graph induced by H 1 + v 1 − v a is connected and v 1 ∈ A h , the graph induced by H is connected. Thus c ( H ) = 1. Note that, for any two vertices u and v in A h − v a , there still exists an hp H (u , v ) since c ( H ) > 1. To prove that H is a hub set, it suﬃces to show that there exists an hp H ( v a , v ) for every vertex v in S \ { v a }. If v is adjacent to v a , then, by deﬁnition, there is an hp H ( v a , v ). It remains to consider the case that v is not adjacent to v a . Clearly, v a is adjacent to some vertex in H 1 + v 1 − v a . Since H 1 + v 1 − v a is connected, there is a path from v a to v 1 such that all intermediate vertices are in H 1 + v 1 − v a . Since v 1 is adjacent to some vertex in H i for each 2 i c ( H ), this implies that an hp H ( v a , v ) exists for every vertex v in S − v a . Therefore, H is a hub set. Note that | H | = | H |. Thus h(G ) = hc (G ) and the lemma follows (Fig. 1). 2 Lemma 2.2. Assume that H is a minimum hub set of a connected graph G and S = V \ H . If N S ( H i ) and N S ( H j ) are disjoint for i = j, then h(G ) = hc (G ), where i , j ∈ {1, 2, . . . , c ( H )} and H 1 , H 2 , . . . , H c ( H ) are the components of the graph induced by H . Proof. Clearly, if c ( H ) = 1, then H itself is a connected set and h(G ) = hc (G ). Thus we assume that c ( H ) > 1. Since N S ( H i ) and N S ( H j ) are disjoint for i = j, by the deﬁnition of hub sets, any vertex in N S ( H i ) for 1 i c ( H ) is adjacent to all other vertices in S \ N S ( H i ). Choose an arbitrary vertex, say v i , from each N S ( H i ) for 1 i c ( H ). It is obvious that v 1 , v 2 , . . . , v c ( H ) form a clique. Let u i be a vertex in H i for 1 i c ( H ) such that H i + v i − u i is still connected. Let H = H ∪ { v 1 , v 2 , . . . , v c ( H ) } \ {u 1 , u 2 , . . . , u c ( H ) } and S = S ∪ {u 1 , u 2 , . . . , u c ( H ) } \ { v 1 , v 2 , . . . , v c ( H ) }. Note that | H | = | H | and c ( H ) = 1. To prove that h(G ) = hc (G ), it suﬃces to prove that H is a hub set. That is, we need to prove that there exists an hp H (x, y ) for every pair of vertices x, y ∈ S . We consider the following cases.

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Fig. 2. A s = V \ H .

Fig. 3. An illustration for Lemma 2.4.

Case 1: x, y ∈ N S ( H i ) ∪ {u i } \ { v i }. If none of x and y is u i , then x and y are adjacent to v j for any j = i. Since v j ∈ H , an hp H (x, y ) exists. If x is u i , then there exists a path from u i to y such that all intermediate vertices are in H i ∪ { v i } \ {u i }. Thus there exists an hp H (x, y ) for this case. Case 2: x ∈ N S ( H i ) ∪ {u i } \ { v i } and y ∈ N S ( H j ) ∪ {u j } \ { v j } for i = j. By using a similar argument as in Case 1, there exists a path from x to v i and a path from y to v j such that all intermediate vertices are in H i ∪ { v i } \ {u i } and H j ∪ { v j } \ {u j }, respectively. Since v i is adjacent to v j , an hp H (x, y ) can be constructed after adding the edge v i v j to the two paths above. Thus this case holds. Case 3: x ∈ N S ( H i ) ∪ {u i } \ { v i } and y ∈ S \

c( H ) i =1

N S ( H i ).

c( H )

Since H is a hub set, it follows that x and y are adjacent when x ∈ N S ( H i ) and y ∈ S \ i =1 N S ( H i ). If x = u i , then there exists a path from u i to v i such that all intermediate vertices are in H i ∪ { v i } \ {u i }. This further implies that an hp H (u i , y ) exists. This completes the proof. 2 Lemma 2.3. Assume that H is a minimum hub set of a connected graph G. If A s = V \ H , then h(G ) = hc (G ). Proof. If A s = V \ H , then A s induces a clique. Since c ( H ) = 1 implies that h(G ) = hc (G ), we assume that c ( H ) > 1 and H 1 , H 2 , . . . , H c ( H ) are the components in H (see Fig. 2). We claim that all N S ( H i ) for 1 i c ( H ) are disjoint. Suppose to the contrary that there exists a vertex v in N S ( H i ) ∩ N S ( H j ) with i = j. Clearly, we can ﬁnd a vertex u i in H i and a vertex u j in H j such that each of H i + v − u i and H j + v − u j is still connected. Let H = H ∪ { v } \ {u i , u j }. It is easy to verify that H is still a hub set with | H | = | H | − 1. This contradicts that H is a minimum hub set of G. Thus N S ( H i ) and N S ( H j ) for i = j are disjoint. By Lemma 2.2, h(G ) = hc (G ). This completes the proof. 2 Lemma 2.4. Assume that H is a minimum hub set of a connected graph G. If A h ∪ A s = V \ H , then h(G ) = hc (G ). Proof. If A s = ∅ or A h = ∅, then, by Lemmas 2.1 and 2.3, this lemma follows. Thus we consider the case where both A h and A s are not empty. Assume that H 1 , H 2 , . . . , H c ( H ) are the components in H with c ( H ) > 1. There exists a vertex, say v p , in A h and a vertex, say v a in H 1 , such that H 1 + v p − v a is connected (see Fig. 3). Let H = H + v p − v a and S = V \ H . Clearly, | H | = | H | and c ( H ) = 1. By using a similar argument as above, it is easy to verify that H is a hub set. Therefore, h(G ) = hc (G ). This completes the proof. 2

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Fig. 4. A trapezoid graph and its corresponding T (G ).

3. Hub sets in co-comparability graphs Deﬁnition 3.1. A graph G = ( V , E ) is a trapezoid graph if there is a trapezoid representation T such that each vertex v i in V corresponds to a trapezoid t i in T and ( v i , v j ) ∈ E if and only if t i and t j intersect in the trapezoid representation. The trapezoid representation consists of two parallel lines L 1 (the upper line) and L 2 (the lower line). A trapezoid t i is deﬁned by its four corner points ai , b i , c i , and di such that [ai , b i ] (respectively, [c i , di ]) is the interval of t i on L 1 (respectively, L 2 ), where ai < bi and c i < di . Therefore, L 1 and L 2 can be labeled with consecutive integers 1, 2, 3, . . . , 2n n n from left to right such that i =1 {ai , b i } = i =1 {c i , di } = [1, 2, . . . , 2n]. Assume that these trapezoids are labeled in increasing order of their corner points b i ’s. We use T (G ) to denote a trapezoid representation of trapezoid graph G. Figs. 4(a) and 4(b) depict a trapezoid graph and its corresponding T (G ), respectively. Trapezoid graphs were introduced by Dagan, Golumbic, and Pinter [3] and, independently, by Corneil and Kamula [2]. Many problems in trapezoid graphs have been studied such as the domination problem [9,11], the coloring problem [3,4], the steiner set problem [11], the vertex-cover problem [12], etc. The connected dominating set problem in trapezoid graphs has also been studied widely [9,11,14]. In [11], Liang presented an O (| V | + | E |)-time algorithm for ﬁnding a connected dominating set with the minimum cardinality in a trapezoid graph G. In [14], Tsai, Lin, and Hsu showed that the minimum weighted connected domination problem in trapezoid graphs can be solved in O (| V | log log | V |) time. In [9], Köhler proposed an O (| V |)-time algorithm for ﬁnding a minimum cardinality connected dominating set in trapezoid graphs. Co-comparability graphs and m-trapezoid graphs are superclasses of trapezoid graphs which are deﬁned as follows. Deﬁnition 3.2. A graph is a co-comparability graph if its complement is a comparability graph. A comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Deﬁnition 3.3. Let L 1 , L 2 , . . . , L m+1 be m + 1 parallel lines, indexed according to their ordering. An m-trapezoid, denoted

+1 m+1 by mi with 1 i n, is an interior of the closed polygon a1i , a2i , . . . , am , bi , . . . , b1i , a1i such that [ai , bi ] is a closed i interval of L j for 1 j m + 1. A graph G = ( V , E ) is called an m-trapezoid graph if and only if there exists an m-trapezoid representation M such that each vertex v i in V corresponds to an m-trapezoid mi in M and ( v i , v j ) ∈ E if and only if mi ∩ m j = ∅. j

j

Lemma 3.4. (See [5].) The class of m-trapezoid graphs is exactly the class of co-comparability graphs of an order P with interval dimension less or equal to m + 1. Lemma 3.5. (See [5].) Given a co-comparability graph G, there exists a positive integer m such that G is an m-trapezoid graph. Lemma 3.6. (See [1].) Co-comparability graphs do not have induced cycles with ﬁve or more edges. We use M (G ) to denote the m-trapezoid representation of an m-trapezoid graph G. Assume that all m-trapezoids in M (G ) are labeled in increasing order of their corner points b1i ’s. In the following theorem, we prove that h(G ) = hc (G ) for m-trapezoid graphs G. Theorem 3.7. If G is a connected m-trapezoid graphs, then h(G ) = hc (G ). Proof. It is obvious that h(G ) hc (G ). Suppose to the contrary that there exists a connected m-trapezoid graph G with h(G ) < hc (G ). Let H be a minimum hub set of G with the least number of components. Clearly, c ( H ) > 1. Let H 1 , H 2 , . . . , H c ( H ) be the components induced by H . The corresponding representation of H i is represented by M ( H i ). Furthermore, we may assume without loss of generality that M ( H i ) is located at the left side of M ( H i +1 ) for 1 i c ( H ) − 1. Let S = V \ H . Clearly, S is not equal to A h ∪ A s ; otherwise, by Lemmas 2.1, 2.3, and 2.4, h(G ) = hc (G ), a contradiction. Thus A o = ∅. Let v k ∈ A o be in as many N S ( H i ) as possible and mk the corresponding m-trapezoid of v k in M (G ). Let N S ( H s ), N S ( H s+1 ), . . . , N S ( H ) with 1 s c ( H ) be those sets containing v k . If s = , then, by the assumption that v k

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is in as many N S ( H i ) as possible, all N S ( H i ) for 1 i c ( H ) are disjoint, and by Lemma 2.2, h(G ) = hc (G ), a contradiction. Thus, in the following, we assume that s < . Let S 1 = N S ( v k ) and S 2 = S \ N S [ v k ]. A vertex v ∈ S 2 is called an s-extreme vertex (respectively, -extreme vertex) if every N S ( H i ) containing v satisﬁes i s (respectively, i ). We claim that if there is an s-extreme vertex in S 2 , then there is no -extreme vertex in S 2 and vice versa. Suppose to the contrary that both s- and -extreme vertices exist in S 2 . Let v s be an s-extreme vertex and v be an -extreme vertex, and ms and m , respectively, be their corresponding m-trapezoids. Since H s and H are two different components and there exists an hp H ( v s , v ), it follows that v s and v must be adjacent. This also reveals that either ms or m must overlap mk in M (G ) which in turn implies that either v s or v is adjacent to v k , a contradiction. Therefore, we may assume without loss of generality that there is no s-extreme vertex with respect to v k . Accordingly, we can ﬁnd a vertex, say v a , in H s such that H s + v k − v a is still connected. Let H = H + v k − v a and S = S + v a − v k . To verify that H is also a hub set, we have to show that there exists an hp H ( v x , v y ) for every pair of nonadjacent vertices v x , v y ∈ S . Case 1: v x , v y ∈ S 1 . In this case, since both v x and v y are adjacent to v k , there exists an hp H ( v x , v y ). Thus this case holds. Case 2: v x , v y ∈ S 2 . By our assumption that there is no s-extreme vertex in S 2 , if S 2 = ∅, then v a is not in any hp H ( v x , v y ). This further implies that any hp H ( v x , v y ) is exactly an hp H ( v x , v y ). Thus this case holds. Case 3: v x ∈ S 1 and v y ∈ S 2 . By the deﬁnition of S 1 , we know that v x is adjacent to v k . Clearly, there is an hp H ( v k , v y ). Since there is no s-extreme vertex in S 2 , vertex v a is not an intermediate vertex in hp H ( v k , v y ). This further implies that all vertices in hp H ( v k , v y ) are in H except v y . Thus there exists an hp H ( v x , v y ) and this case holds. Case 4: v x = v a and v y ∈ S 1 ∪ S 2 . Since v a is adjacent to some vertex in H s + v k − v a , there exists an hp H ( v a , v k ). By using similar arguments as in Cases 2 and 3, it follows that hp H ( v a , v y ) exists. From above, we can ﬁnd that H is also a hub set. It is easy to verify that | H | = | H | and c ( H ) < c ( H ). This contradicts that H is a minimum hub set with the least number of components. This establishes the proof. 2 In Theorem 3.8, we describe the main result in [8] by Johnson, Slater, and Walsh. Then, in Theorem 3.10, we discuss the relationship between hc (G ) and γc (G ) in co-comparability graphs. Theorem 3.8. (See [8].) Suppose that G is connected. Then, γc (G ) = hc (G ) + 1 if and only if one of the following holds: (i) G is a clique. (ii) For some vertex u ∈ V , T = V \ N [u ] = ∅ induces a clique in G, T and N (u ) are the parts of a complete bipartite subgraph of G, and no vertex in N (u ) has degree | V | − 1. (iii) Some set S ⊆ V with | S | 2 induces a path in G; T = V \ N [ S ] = ∅ induces a clique in G, and T and U = N ( S ) \ S are the parts of a complete bipartite subgraph of G. No vertex of S other than the end-vertices u , v of the path G [ S ] has neighbors in U , and the sets A = N (u ) ∩ U and B = N ( v ) ∩ U are nonempty and disjoint. (a) If | S | = 2, then no vertex of A dominates B, no vertex of B dominates A, and for any edge ab ∈ E, a ∈ A, b ∈ B, {a, b} does not dominate A ∪ B. (b) If | S | = 3, then for any edge ab ∈ E, a ∈ A, b ∈ B, {a, b} dominates neither A nor B. (c) If | S | 4, then there are no edges between A and B. According to Theorem 3.8, a graph is called a Type I, II, or III graph if it satisﬁes Condition (i), (ii), or (iii), respectively, of Theorem 3.8 (see Fig. 5). Obviously, it can be done in O (| V |) time to recognize whether a graph is a Type I graph provided that the degree of each vertex is given; otherwise, computing the degrees for all vertices can be done in O (| V | + | E |) time. In the rest of this paper, we assume that the input contains the degree of each vertex. We introduce a new data structure, called mixed adjacency lists, to represent graphs. We will use this data structure to represent a graph G. In a mixed adjacency list, every vertex is associated with an adjacency list or a nonadjacency list as well as its degree. If the degree of vertex v, denoted by d( v ), is less than or equal to n/2, then the list associated with v is its original adjacency list; otherwise, the list associated with v is the nonadjacency list of v, namely the vertices in V \ N [ v ]. Example 1. In Fig. 6(a), there is a graph of 8 vertices. Fig. 6(b) shows the mixed adjacency list of the graph. We can ﬁnd that the degree of vertex a is 6 which is greater than n/2 = 4. Note that V \ N [a] = {h}. Thus the list associated with vertex

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Fig. 5. Type II and III graphs.

Fig. 6. A graph and its mixed adjacency list.

a contains only one vertex h. However, the degree of vertex h is 3 which is less than or equal to n/2 = 4. Thus the list associated with vertex h contains vertices e, f , and g which are the vertices in its original adjacency list of h. It is easy to verify that the length of an associated list in mixed adjacency list is at most n/2. For some classes of graphs, e.g., complete graphs and split graphs, using mixed adjacency list can save a lot of memory space. Lemma 3.9. By using the mixed adjacency list, recognizing Type II graphs can be done in O (| V |) time. Proof. Let n = | V |. By the deﬁnition of Type II graphs, a graph is a Type II graph if and only if the following conditions are satisﬁed: (1) (G ) = n − 2, (2) d( v ) = n − 2 for every v ∈ T , and (3) there is at least a vertex u with d(u ) = n − | T | − 1, where T = V \ N [u ] = ∅ and (G ) is the maximum degree of G. If d( v ) = n − 2, then there is only one vertex, say w, in V which is not adjacent to v. We call w the missing vertex of v. Let M w denote the set containing all vertices whose missing vertex is w. By examining the degree of every vertex in V , we can ﬁnd all vertices of degree n − 2 in O (n) time. Accordingly, all missing vertices w and M w can be found in O (n) time. To recognize a Type II graph is equivalent to determining whether there is a missing vertex w with d( w ) = n − | M w | − 1. Clearly, n − | M w | − 1 can be computed in O (n) time for all missing vertices. Therefore, it can be done in O (n) time to determine whether a graph is a Type II graph. This completes the proof. 2 Example 2. In Fig. 6(a), we can ﬁnd that vertices e and h are missing vertices. Furthermore, M e = {∅} and M h = {a, b, c , d}. It is easy to check that d(h) = n − | M h | − 1 = 8 − 4 − 1 = 3 while d(e ) = n − | M e | − 1 = 8 − 0 − 1 = 7. The graph in Fig. 6(a) is a Type II graph.

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Fig. 7. An illustration for Theorem 3.10.

Theorem 3.10. For a co-comparability graph G, if G is a Type I or II graph, then hc (G ) = γc (G ) − 1; otherwise, hc (G ) = γc (G ). Proof. We claim that no co-comparability graph is a Type III graph. For simplicity, we follow the terms deﬁned in Condition (iii) of Theorem 3.8. By Condition (iii) of Theorem 3.8, there exist two vertices c ∈ A and d ∈ B such that c is not adjacent to d. Since T = φ , there exists a vertex e ∈ T such that edges ce ∈ E and de ∈ E. Since A = N (u ) ∩ U and B = N ( v ) ∩ U are nonempty and disjoint, vertices u, c, e, d, v, and the path from u to v in S form an induced subgraph isomorphic to C k with k 5 (see Fig. 7). By Lemma 3.6, G is not a co-comparability graph. This claim holds. By Theorem 3.7 and the inequality h(G ) hc (G ) γc (G ) h(G ) + 1 in [6], we can derive that γc (G ) − 1 hc (G ) γc (G ) for co-comparability graphs G. This completes the proof. 2 Theorem 3.11. By using the mixed adjacency list, h(G ) and hc (G ) of a trapezoid graph G can be computed in O (| V |) time. Proof. Recognizing whether G is a complete graph can be done in O (| V |) time. By Lemma 3.9, it can be done in O (| V |) time to recognize a Type II graph. By the results in [9,14], a minimum connected dominating set of a trapezoid graph G can be found in O (| V |) time. Thus, by Theorems 3.7 and 3.10, h(G ) and hc (G ) can be computed in O (| V |) time. This completes the proof. 2 Theorem 3.12. The h(G ) and hc (G ) of a co-comparability graph G can be computed in O (n3 ) time. Proof. By the results in [10], a minimum connected dominating set of a co-comparability graph G can be found in O (n3 ) time. This completes the proof. 2 4. Concluding remarks In this paper, we investigate the hub number of co-comparability graphs. We prove that h(G ) = hc (G ) and characterize the case for which γc (G ) = hc (G ) of co-comparability graphs. Furthermore, we show that h(G ) and hc (G ) of a trapezoid graph G can be determined in O (n) time and h(G ) and hc (G ) of a connected co-comparability graph G can be determined in O (n3 ) time. As a further study, it is interesting to investigate whether h(G ) = hc (G ) holds in perfect graphs and AT-free graphs. References [1] H. Breu, Algorithmic aspects of constrained unit dis graphs, PhD thesis, University of British Columbia, 1996. [2] D. Corneil, P. Kamula, Extensions of permutation and interval graphs, in: Proceedings of the 18th Southeast, Conference on Combinatorics, Graph, and Computing, Boca Raton, FL, in: Congr. Numer., vol. 58, 1987, pp. 267–275. [3] I. Dagan, M.C. Golumbic, R.Y. Pinter, Trapezoid graphs and their coloring, Discrete Appl. Math. 21 (1988) 35–46. [4] S. Felsner, R. Muller, L. Wernisch, Trapezoid graphs and generalizations, geometry and algorithms, Discrete Appl. Math. 74 (1997) 13–32. [5] Carsten Flotow, On powers of m-trapezoid graphs (Note), Discrete Appl. Math. 63 (1995) 187–192. [6] T. Grauman, S.G. Hartke, A. Jobson, B. Kinnersley, D.B. West, L. Wiglesworth, P. Worah, H. Wu, The hub number of a graph, Inform. Process. Lett. 108 (2008) 226–228. [7] P. Hamburger, R. Vandell, M. Walsh, Routing sets in the integer lattice, Discrete Appl. Math. 155 (2007) 1384–1394. [8] P. Johnson, P. Slater, M. Walsh, The connected hub number and the connected domination number, Networks 58 (2011) 232–237. [9] E. Köhler, Connected domination and dominating clique in trapezoid graphs, Discrete Appl. Math. 99 (2000) 91–110. [10] Dieter Kratsch, Lorna Stewart, Domination on cocomparability graphs, SIAM J. Discrete Math. 6 (1993) 400–417. [11] Y.D. Liang, Steiner set and connected domination in trapezoid graphs, Inform. Process. Lett. 56 (1995) 101–108. [12] M.S. Lin, Y.J. Chen, Counting the number of vertex covers in a trapezoid graph, Inform. Process. Lett. 109 (2009) 1187–1192. ´ [13] C.H. Lin, J.J. Liu, Y.L. Wang, W.C.K. Yen, The hub number of Sierpinski-like graphs, Theory Comput. Syst. 49 (3) (2011) 588–600. [14] Y. Tsai, Y. Lin, F.R. Hsu, Eﬃcient algorithms for the minimum connected domination on trapezoid graphs, Inform. Sci. 177 (2007) 2405–2417. [15] M. Walsh, The hub number of a graph, Int. J. Math. Comput. Sci. 1 (2006) 117–124.

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