Algorithm for the Vertex Connectivity Problem on Circular Trapezoid Graphs

The vertex connectivity ( ) G κ of a graph G is the minimum number of nodes whose deletion disconnects it. Graph connectivity is one of the most fundamental problems in graph theory. In this paper, we designed an ( ) 2 O n time algorithm to solve connectivity problem on circular trapezoid graphs.


Introduction
The vertex connectivity ( ) G κ of a graph G is the minimum number of nodes whose deletion disconnects it. The computation of ( ) G κ for a given graph G is known as the vertex connectivity (vertex connectivity) problem, and it is one of the most fundamental problems in graph theory. In recent years, many studies related to vertex connectivity have been conducted [1]- [6]. Even and Tarjan developed an ( ) 1.5 O mn time algorithm to calculate the vertex connectivity of a general graph [7]. In many cases, more efficient algorithms can be developed by restricting the classes of graphs. For example, Ghosh and M. Pal presented an ( ) 2 O n time algorithm to solve the VC problem for trapezoid graphs [8]. Subsequently, this algorithm was improved by Ilić [9] to ( ) log O n n time by using a binary indexed tree.
Lin introduced circular trapezoid graphs (CTG), which constitute a proper superclass of trapezoid graphs and circular-arc graphs [10]. He  to solve the VC problem on CTGs. Our algorithm was realized by skillfully combining the methods of [9] and [11]. The rest of this paper is organized as follows. Section 2 describes some definitions of circle trapezoid graphs and models and introduces the extended circle trapezoid model, as well as some notations. Section 3 presents some properties on circle trapezoid graphs, which are useful for finding vertex connectivity in an efficient manner. Section 4 describes our algorithm for the VC problem and its complexity. Finally, Section 5 concludes the paper.

Definitions
We describe the circular trapezoid model (CTM) before defining the CTG. The model comprises inner and outer circles C 1 and C 2 with radii 1 2 r r < , respectively. Each circle is arranged counterclockwise with consecutive integer values 1, 2, , 2n  , where n is the number of trapezoids. Consider the two arcs, A 1 and A 2 , on C 1 and C 2 , respectively. Points a and b are the first points encountered when traversing the arc A 1 counterclockwise and clockwise, respectively; similarly, points c and d are the first points encountered when traversing the arc A 2 counterclockwise and clockwise, respectively. A trapezoid is the region in circles C 1 and C 2 that lies between two non-crossing chords ac and bd. A trapezoid illustrates an example of a CTM M having 8 trapezoids. For example, CTM is used for cities comprising cityscapes that spread radially around facilities such as stations and rotaries. It is used to visually represent the relationships among communities (linkage of transportation networks, sharing of infrastructure facilities, etc.), and it is applied to the optimization of city planning and facility arrangement. Table 1 shows the details of M as depicted in Figure 1  A graph G is a CTG if it can be represented by the following CTM M: each vertex of the graph corresponds to a trapezoid, and two vertices in G are considered adjacent if and only if their corresponding trapezoids intersect. Figure 1(b) illustrates the CTG G corresponding to CTM M shown in Figure 1(a). In this example, G is disconnected by removing vertices 1, 5, and 6 from G. Thus, the vertex connectivity of G is 3.
In the following, we introduce an extended circular trapezoid model (ECTM) constructed from a CTM. Let n be the number of trapezoids in CTM M. Consider a fictitious line p that connects the points placed between 1 and 2n of C 1 and C 2 . First, we cut CTM along fictitious line p and expand the two circles C 1 and C 2 into parallel horizontal lines called top and bottom channels, respectively.
Hereafter, to avoid confusion, we denote trapezoids in CTM and ECTM by i CT and i T , respectively. Finally, for each i T , 1 i n ≤ ≤ , copies of i n T + and i n T − are created by shifting 2n to the right and left, respectively. An ECTM is constructed from a CTM by the above process, which can be executed in ( ) O n time [11]. Figure 2 illustrates an ECTM EM constructed from the CTM M shown in Figure 1(a). Table 2 shows the details of EM.
Some notations that form the basis of our algorithm in Section 4 are defined as follows. A separating set in a connected graph G is a set of vertices whose deletion disconnects G. We introduce a new concept to ECTM that is similar to the separating set in CTG. A separating trapezoid set in an ECTM EM is a set of trapezoids whose deletion separates EM into two or more components. Let S be a separating trapezoid set of EM. EM S − is a trapezoid set that is obtained by deleting S from all trapezoid sets of EM. If EM S − has k components, we de-

Properties of Vertex Connectivity on TCGs
We describe some lemmas that are useful for constructing the algorithm for the VC problem.

Outline of Algorithm
Efficient algorithms that address various problems concerning non-circular intersection graphs (interval, permutation, trapezoid, etc.) have been developed. However, in general, problems for circular intersection graphs tend to be more difficult than those for non-circular intersection graphs. One cause is because, in contrast to non-circular intersection graphs, we cannot determine the starting position of an algorithm uniquely for a circular intersection graph owing to the existence of feedback elements. For several problems, we can develop circular versions of the existing algorithms by constructing extended intersection models for the problems. By using extended intersection models such as an ECTM, we can determine the start position of an algorithm uniquely and apply the algorithms of the non-circular versions partially. For instance, this method has been applied to develop efficient algorithms for the shortest path query problem [12] [13] and the articulation vertex problem [14] on circular-arc graphs, maximum clique and chromatic number problems [15], the spanning forest problem [16] and the articulation problem [17] on circular permutation graphs, and the spanning tree problem [11] and the hinge vertex problem [18] on circular trapezoid graphs.
Here, we concisely describe the outline of our algorithm. When a given CTG has articulation vertices, the vertex connectivity is 1. We can find articulation vertices in

( )
O n m + time by applying the traditional method with depth first search. Then, we discuss graphs that do not contain articulation vertices.  Figure 4). Again, we compute the minimum cardinality separating trapezoid sets i S′ of EM ′ by using Ilić's algorithm [9]. In the example of

Algorithm VC-CTG and Its Analysis
In this section, we present Algorithm VC-CTG to compute the vertex connectivity of a CTG G. We formally describe Algorithm VC-CTG as follows. A CTM M is taken as an input. Our algorithm uses both Ilić's and Honma et al.'s algorithms [9] [11].  Here, we analyze the complexity of Algorithm VC-CTG. In Step 1, we check whether given graph G has articulation vertices using the traditional algorithm.

Conclusion
In this study, we proposed Algorithm VC-CTG, which operates in we believe that this paper is significant from both theoretical and algorithmic perspectives. Future research will address reducing the complexity of the algorithm and extending the results to other graphs.