Abstract
The crossing number of graph is the minimum number of edges crossing in any drawing of in a plane. In this paper we describe a method of finding the bound of 2-page fixed linear crossing number of . We consider a conflict graph of . Then, instead of minimizing the crossing number of , we show that it is equivalent to maximize the weight of a cut of . We formulate the original problem into the MAXCUT problem. We consider a semidefinite relaxation of the MAXCUT problem. An example of a case where is hypercube is explicitly shown to obtain an upper bound. The numerical results confirm the effectiveness of the approximation.
1. Introduction
Let be a simple connected graph with a vertex-set and an edge-set . The crossing number of graph , denoted , is the minimum number of pairwise intersections of edge crossing on the plane drawing of graph . Clearly, if and only if is planar. It is known that the exact crossing numbers of any graphs are very difficult to compute. In 1973, Erdös and Guy [1] wrote, “Almost all questions that one can ask about crossing numbers remain unsolved.” In fact, Garey and Johnson [2] prove that computing the crossing number is NP-complete.
A 2-page drawing of is a representation of on the plane such that its vertices are placed on a straight horizontal line according to fixed vertex ordering and its edges are drawn as a semicircle above or below but never cross .
The -cube or -dimensional hypercube is recursively defined in terms of the Cartesian products. The one-dimension cube is simply where is a complete graph with vertices. For , is defined recursively as . The order of is and its size is . Since is planar for , so for each such . Eggleton and Guy [3] showed that but is unknown for .
It was declared by Eggleton and Guy [3] that the crossing numbers of the hypercube (non-2-page) for was
Then, in 1973, Erdös and Guy [1] conjectured equality in (1). In 1993, a lower bound of was proved by Sýkora and Vrt’o [4]:
In 2008, Faria et al. [5] constructed a new drawing of in the plane which led to the conjectured number of crossings To the best of our knowledge, the fixed linear crossing number for has not been established. In this paper, we discuss a method to obtain an approximation for fixed linear crossing number for hypercube graph.
2. 2-Page Drawings of Hypercube Graph
Throughout this paper, we consider the ordering of hypercube graph . Since is defined recursively as , for , where is a simple graph with 2 vertices together with a single edge incident to both vertices, has 2 copies of with edges connecting between them. Given a fixed ordering on , the vertices of the first are labeled and the vertices of the second are labeled . The two vertices between the first and the second are adjacent if and only if the sum of the labeled is . Figures 1 and 2 present the ordering of and which we consider throughout this paper. Notice that our method is independent on vertex ordering; therefore, for a fixed , we can apply the method times so as to obtain the 2-page linear crossing number.
The 2-page drawing of can be represented by drawing the vertices of on a straight horizontal line according a fixed vertex ordering. Each edge fully contained one of the two half-planes (pages) as a semicircle and never cross . Notice that no edge crosses itself, no adjacent edges cross each other, no two edges cross more than once, and no three edges cross in a point.
For a given 2-page drawing of with the fixed vertex ordering, a pair of edges and are potential crossing if and cross each other when routed on the same side of . Clearly, and are potential crossing if and only if or .
Next we give the definition of conflict graph of graph .
Definition 1. Given a graph . We define an associated conflict graph of a graph . There is corresponding one-to-one and onto mapping between the set of and . Two vertices of are adjacent if any two edges in are potential crossing.
For example, according to the given fixed vertex ordering of (see Figure 3), is a graph of nodes, . and are adjacent in because and are potential crossing in a 2-page drawing of . A fixed vertex ordering of and its potential crossing can be seen in Figure 4.
In this paper, we are interested only in fixed linear embeddings of . There is a crossing between and if and only if and are potential crossing and embedded on the same side of . We can see that the number of edge crossings depends on the order of vertices and on the sides to which the edges are assigned.
The 2-page linear crossing number of , denoted by , is the minimum number of pairwise intersections of edges crossings determined by a 2-page drawing of . The 2-page fixed linear crossing number of is the minimum number of pairwise intersections of edges crossings determined by a 2-page drawing of with fixed vertex ordering of . It is known that for , for
3. Reduction to MAXCUT Problem
In this section, we show that the problem can be reduced to the maximum cut problem. Next, we reduce the fixed linear crossing number problem to the maximum cut problem (MAXCUT). The MAXCUT problem is as follows.
Maximum Cut Problem (MAXCUT). Given an undirected graph the edge of the graph is associated with nonnegative weights . The problem is to find a cut of the largest possible weight, that is, to partition the set of into disjoint sets and such that the total weight of all edges linking and (i.e., with one incident node in and the other one in ) is as large as possible.
In the MAXCUT problem, we may assume that the weights are defined for every pair of indices: it suffices to set for pairs of nonadjacent nodes. For the unweighted graph, we assume that for .
Let be a graph with a fixed vertex permutation. Given a vertex partition of its conflict graph , the associated cut embedding is the fixed linear embedding of where edges corresponding to and are embedded to the half spaces above and below the vertex line, respectively.
Lemma 2 (see [6]). where is a number of potential crossing of 2-page drawing of , which is the number of edges of . is the size of the maxcut of .
Proof. Given a 2-page (circle) drawing of , define as the chords that are drawn inside the circle. The edges of with precisely one endpoint in now correspond to edges of that do not cross in the drawing.
Theorem 3 (see [7]). Consider a partition of . Then the corresponding cut embedding is a fixed linear embedding of with a minimum number of crossings if and only if is a maximum cut in .
Proof. Let be the set of edges in with one endpoint in and one endpoint in , that is, the cut given by . By definition of , we know that every crossing in the cut embedding associated with corresponds to an edge in such that either both its endpoint belong to or both belong to , that is, to an edge in . Thus, the number of crossings is . As is constant for a fixed vertex permutation, the result follows.
Theorem 3 reduces the fixed linear crossing number problem to the maximum cut problem (MAXCUT). In the next section, we describe the relaxation of the MAXCUT problem which leads to semidefinite programming.
3.1. Formulating MAXCUT by Semidefinite Relaxation
In this section, we show that 2-page crossing number of hypercube graph problem can be obtained by computing a semidefinite relaxation of MAXCUT.
First of all, we introduce the adjacency matrix of denoted as we know it is an matrix with the property
From we construct the conflict graph of denoted . Finally, we perform MAXCUT on graph . We use semidefinite relaxation to approximate the optimal value solution to the MAXCUT problem. Obviously the approximation is larger than the actual MAXCUT optimal value. The feasibility of the relaxation set is strictly larger than the original ones.
According to [2], the MAXCUT problem can be formulated as follows:We call the optimal value of (6) as “OPT.” Then, the relaxation of (6) can be rewritten aswhere is an adjacency matrix of and is a feasible solution to the semidefinite relaxation. The problem (7) is equivalent towhere is a given adjacency matrix of and is a feasible solution to the semidefinite relaxation. We call the optimal value of (8) as “SDP.”
As we have seen from the relation (4), we let be the number of potential crossing of 2-page drawing of with our fixed vertex ordering (i.e., is the number of edges of ).
We can determine by considering the upper half of the main diagonal of the adjacency matrix of .
Definition 4. Let be the adjacency matrix of . The element where is called minor diagonal of adjacency matrix of and the element where is called semiminor diagonal of adjacency matrix of , denoted by .
For simplicity, the size of is a number of elements in . Let be the adjacency matrix of graph . Therefore is symmetric matrix. It is clear that the size of is .
Let and be adjacency matrices of graphs and , respectively; we say that the number of potential crossing between and , denoted by , is simply the number of potential crossing between 2-page drawing of graph and . The adjacency matrix of size of with respect to our ordering is presented in Figure 5.
Lemma 5. For any integer ,where .
Lemma 6. For every adjacency matrix of , , where , there exists adjacency matrix of , , which is a submatrix embedding in . The number of submatrix embedding in is .
Lemma 7. For any integer ,where .
Lemma 8. For any integer ,where .
Lemma 9. For , the number of block embedding in equals , .
Lemmas 5–9 follow directly from the definitions.
Lemma 10. For any integer ,
Proof. The number of potential crossing between and is a result of the number of potential crossing between all of the element in and . That is,
Theorem 11. For any integer ,where is the number of potential crossing of 2-page drawing of with our fixed vertex ordering.
Proof. We prove this lemma by considering the number of potential crossing of 2-page drawing of with our fixed vertex ordering. Since has copies of with some edges connecting between them, the number of potential crossing of is a result of twice of the number of potential crossing within together with and .
Hence, it is enough to show that the number of potential crossing between 2-page drawing of and is equal to , whereNote is the number of potential crossing between all of submatrices and and also between and . By Lemmas 6 and 9,We precede by mathematical induction on . For , it can be easily seen that by counting. Assuming (15) holds true, now we consider as a number of potential crossing between all of the submatrices and and also between and . By the Lemmas 5, 6, 7, 9, 8, and 10,
The next theorem shows how effective the relaxation is.
Theorem 12 (see [8]). Let OPT be the optimal value of the MAXCUT problem and SDP be the optimal value of the semidefinite relaxation. Then
Theorem 12 guarantees that the optimal value of the MAXCUT is close to the optimal value of the semidefinite relaxation. From (4), we havewhere is a number of potential crossing of 2-page drawing of . is an approximation of 2-page fixed linear crossing number of and is an approximation of .
Corollary 13. Let be an approximation of . Then we havewhere is a computable quantity depending on .
Proof. From (4), (18), and (19), we have Let be the computable quantity depending on .
Then,
Corollary 13 shows that the upper bound of is , where is the computable quantity depending on .
3.2. Experimental Results
In this section, we consider the hypercube graph for . Then, we give some examples for approximating the problems of the semidefinite relaxation in the form (8). We approximate this problem via MATLAB program together with an optimization toolbox called “SeDuMi.” The SeDuMi is a package for solving optimization problems with linear, quadratic, and semidefinite constraints.
In Table 1, the second column shows numerical results for the approximation of the MAXCUT on the associated conflict graph by using the semidefinite relaxation. It is well known that this problem can be solved in a polynomial time. The third column displays the numbers of potential crossing of 2-page drawing of referring to our fixed vertex ordering that we evaluate from (14). Notice that this potential crossing of 2-page drawing of is the exact value. From (19), we calculate the approximation of 2-page fixed linear crossing number of for . The results are shown in the last column.
In Table 2, we present the lower bound of , and the upper bound of , . The second column shows the values of for . We see that as get larger the values of tend to decline continuously. It is interesting to study the behavior of as . It does not surprise to see that our approximation is strictly larger than the upper bound of (14) since the latter one does not have a restriction that all vertices must be placed on a line. However, it is surprising to see that these numbers are not so different from each other.
4. Concluding Remarks
In this paper, given graph , we show how the associating conflict graph is constructed. We recharacterize the problem of finding the crossing number of graph to the MAXCUT problem of . We approximate the MAXCUT problem by the semidefinite relaxation which can be solved easily by a standard optimization package; in this case, we use SeDuMi 1.02. The numerical results show reasonable outcome. Clearly, another relaxation method can be explored. Moreover, it would be quite interesting to see the behavior of as get larger. One can further study how to estimate for a larger .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors would like to thank the Thailand Research Fund under Project RTA5780007 and Chiang Mai University, Chiang Mai, Thailand, for the financial support.