Improved Bounds for the Crossing Number of the Mesh of Trees

Transcription

1 Improved Bounds for the Crossing Number of the Mesh of Trees ROBERT CIMIKOWSKI Computer Science Department Montana State University Bozeman, Montana , USA IMRICH VRT O Institute for Informatics Slovak Academy of Sciences Bratislava, Slovak Republic vrto@savba.sk Abstract Improved bounds for the crossing number of the mesh of trees graph, M n, are derived. In particular, we derive a new lower bound of 5n 2 log n 44n 2 80 which improves on the previous bound of Leighton [11] by a constant factor, and an upper bound of (log n 10 3 )n2 + 8n In addition, we construct drawings of M n which achieve the upper bound number of crossings. We also prove that the crossing number of M 4 is 4. Key words. crossing number, mesh of trees, parallel architecture, VLSI layout. All logarithms in this paper are base 2. 1

2 Crossing Number Bounds for the Mesh of Trees 2 1 Introduction The mesh of trees graph has been studied as a model for parallel computer architectures (see [12]). The vertices of the graph represent processors and the edges denote the communication channels between the processors. Due to its small diameter and large bisection width, the mesh of trees is currently the fastest network known for problems such as matrix-vector multiplication and sorting when considered solely in terms of computational speed. When fabricating a VLSI circuit for a network, it is critical to minimize the amount of area required for the network layout. In [2, 11] it is shown that the crossing number of a graph strongly influences the layout area required. In this paper we determine improved lower bounds and an upper bound for the crossing number of the mesh of trees graph. To the authors knowledge, no previous nontrivial upper bounds have been published. A lower bound was previously given in [11] but its value is positive only for very large instances. We derive a new lower bound by improving upon the constants in [11] and show that its values are significant for much smaller instances of the graph. We also construct recursive drawings of M n which achieve the number of crossings given by the upper bound. Here, we will assume that a graph G = (V, E) is finite, simple, and undirected. The crossing number of G, ν(g), is the minimum number of edge crossings necessary in any drawing of G in the plane. Exact crossing numbers have been determined only for small instances of special families of graphs, e.g., complete and complete bipartite graphs [4, 9]. Some recent results and surveys of the crossing number problem may be found in [14, 15, 16]. For arbitrary G, computing ν(g) is NP-hard [6], and therefore likely to be intractable. Hence, it is more practical to explore bounds for this parameter. Such bounds have previously been given for two other popular parallel network topologies the hypercube [13, 20, 5] and cube connected cycles [20]. Also, in [1, 3, 10, 17], exact crossing numbers for special cases of another parallel network model, the 2-dimensional mesh, are given. 2 Mesh of Trees The n n mesh of trees graph, M n, where n is a power of 2, is constructed from an n n grid of vertices by adding extra vertices and edges to form a complete binary tree in each row and column. The extra vertices are the

3 Crossing Number Bounds for the Mesh of Trees 3 Figure 1: 4 4 mesh of trees M 4. internal vertices of the trees, and the i th row and j th column trees share a unique leaf in position (i, j) of the grid. The trees are constructed so that the leaves in each tree are precisely the vertices in the corresponding row or column of the original n n grid, and the subgraph induced on the vertices in each quadrant is M n/2. M n has 3n 2 2n vertices and 4n 2 4n edges. The quadrants are defined by two perpendicular axes drawn through the roots of the row and column trees. M 4 is shown in Figure 1. 3 Nonplanarity of M n Although M 2 is planar, we show that M n is nonplanar for n 4. Edges between tree roots in interior rows and columns obstruct planarity. Define a contraction as the operation of replacing an edge e = uv of a graph with a single vertex, denoted by uv, adjacent to all neighbors of u and v with multiple edges deleted. We make use of the following variant of Kuratowski s Theorem: Lemma 3.1. [7, 21]. A graph is planar iff it does not contain a subgraph contractible to K 5 or K 3,3.

4 Crossing Number Bounds for the Mesh of Trees 4 Lemma 3.2. M n is contractible to K n,n. Proof: (By induction on n.) For the basis, we show that M 4 is contractible to K 4,4. Initially, we suppress any degree-2 vertices of M 4 since this operation simplifies the graph without affecting the planarity or nonplanarity (see Figure 2a). Next we apply a series of edge contractions, as shown in Figure 2b-c, to obtain the minor K 4,4. In general, vertex label uv denotes the contracted edge uv. Now assume the lemma is true for n k, where k is a power of 2, k 4, and consider M 2k. M 2k contains four M k subgraphs, each of which contracts to K k,k. Define a link edge as an edge from one M k subgraph to another M k subgraph. There are 4k link edges, 2k horizontal and 2k vertical, interconnecting the four subgraphs. The arrangement for M 8 is shown in Figure 3. Horizontal edges u i v i, 1 i 2k, are contracted to single vertices u i v i to form 2k vertices < u 1 v 1 >,..., < u 2k v 2k >. The vertical link edges x i y i, 1 i 2k, are similarly contracted to single vertices x i y i to form 2k vertices < x 1 y 1 >,..., < x 2k y 2k >. Since each of u i and v i, after contraction, have common neighbors in M 2k, each vertex u i v i is adjacent to all of the vertices < x 1 y 1 >,..., < x 2k y 2k >, and the vertices < u 1 v 1 >,..., < u 2k v 2k > are mutually nonadjacent. The same is true for the vertices x i y i and < x 1 y 1 >,..., < x 2k y 2k >. Hence, the resulting adjacency structure is that of K 2k,2k, and the lemma follows. Lemma 3.2 is actually stronger than needed; nevertheless, the following theorem is immediate. Theorem 3.3. M n is nonplanar for n 4. 4 The Crossing Number of M 4 Since degree-2 vertices do not affect the planarity or crossing number of a graph, henceforth we consider only suppressed drawings of M 4, such as that of Figure 2a. Let D be a suppressed drawing of M 4 in the plane. A cycle C is facial in D if it does not intersect itself and its interior contains no vertices or edges. For example, the four 4-cycles of Figure 2a are all facial, while none of the four 6-cycles are. D is 4-facial if all four of its 4-cycles are facial.

5 Crossing Number Bounds for the Mesh of Trees 5 a 1 b 1 a 1 b 1 a 4 a 2 a 3 b 4 b 2 b 3 d 1 c 1 a 4 a 2 a 3 b 3 d 1 c 1 b 4 b 2 d 4 d 2 c 4 c 2 d d 2 c 4 4 c 2 d 3 c 3 d 3 c 3 (a) (b) a 1 b 1 a 4 d 4 a 3 b 3 a 2 d 2 b 4 c 4 b 2 c 2 d 1 c 1 d 3 c 3 (c) Figure 2: Illustration of proof of Lemma 3.2: (a) M 4 with degree-2 vertices suppressed and (b-c) after further edge contractions reduce it to K 4,4.

6 Crossing Number Bounds for the Mesh of Trees 6 Figure 3: M 8 contracted to four interconnected copies of K 4,4. We next partition the set of all drawings of M 4 into two subsets those in which some edge crosses a 4-cycle, and those in which no edge crosses a 4-cycle. The next two lemmas bound the number of crossings in either type of drawing. Lemma 4.1. If D is a drawing of M 4 in which no 4-cycle has an edge crossed, then D has at least 4 crossings. Proof: First, we note that each 4-cycle of M 4 (with degree-2 vertices suppressed) is joined by 2 pairs of link edges to 2 other 4-cycles. Then there are two cases to consider: either D is 4-facial as shown in Figure 2a, or else one of the 4-cycles contains the other three facial 4-cycles. Any other arrangement of the 4-cycles must have one of their edges crossed, since there would be two 4-cycles A and B separated by another 4-cycle C and the link edges between A and B would cross C. case 1: Assume the 4-cycles are all facial and refer to the drawing of M 4

7 Crossing Number Bounds for the Mesh of Trees 7 in Figure 2a. Let the 4-cycles be C 1,..., C 4 in clockwise order starting with the upper left quadrant of the figure. Each C i is joined by two link edges to C (i mod 4)+1, 1 i 4. Let H 1 denote the subgraph induced by the vertices of C 1 and C 2, and let H 2 be the subgraph induced by the vertices of C 3 and C 4. Then the two edges from interior vertices a 2, b 4 of H 1 to interior vertices d 2, c 4 of H 2 must either cross the cycle (a 1, b 1, b 2, b 3, a 3, a 4, a 1 ) or the edge a 3 b 3, or else edges a 3 b 3 and d 1 c 1 must incur 4 crossings if redrawn to intersect edges a 4 d 4 and b 2 c 2, respectively, and then continue through the exterior face of D. Similarly, either the same pair of edges must cross the cycle (d 1, c 1, c 2, c 3, d 3, d 4, d 1 ) or else edge d 1 c 1 must cross the exterior cycle of D twice, resulting in a total of 4 crossings. By the same argument, edges a 3 b 3 and d 1 c 1 must incur 4 crossings whether each is drawn inside or outside of the cycle along the exterior face of D. Hence, there are at least 4 crossings in D. case 2: Assume, wlog, that 4-cycle C 1 contains 4-cycles C 2, C 3, C 4. We examine the incremental embedding of M 4 in four steps, with step i corresponding to the embedding of cycle C i. At step 2, C 2 is embedded inside C 1 and has two link edges to vertices of C 1 (Figure 4a). The link edges and C 2 divide the interior of C 1 into two regions r 1 and r 2. At step 3, C 3 cannot be embedded inside C 2 or the link edges of C 3 or C 4 would cross C 2 or C 3 depending on whether C 4 is embedded inside C 2 or C 3. Hence, C 3 must be embedded inside r 1 or r 2, and one of its link edges must cross a link edge from C 2 to C 1, which results in the formation of 4 interior regions r 1, r 2, r 3, r 4 (Figure 4b). Finally, at step 4, C 4 must be embedded inside one of the four regions. Regardless of which region C 4 is embedded in, one pair of its link edges must join with C 3, and this adds a second crossing. Of the remaining two link edges to C 1, there are two subcases to consider: (1) one causes a third crossing with a link edge from C 2 to C 1, and the other must cause a fourth crossing with a link edge from C 4 to C 3 (Figure 4c); or (2) one link edge causes two crossings, the third and fourth, with a link edge from C 4 to C 3. Lemma 4.2. If D is a drawing of M 4 in which some 4-cycle has an edge crossed, then D has at least four crossings. Proof: There are three cases to consider either two or more 4-cycles intersect, or else they do not intersect and either one or two 4-cycles have an

8 Crossing Number Bounds for the Mesh of Trees 8 r 2 C 1 r 1 C 2 C 1 r 2 r 1 C 2 C 3 r 3 r 4 (a) (b) C 1 r 2 r 1 C 2 C 3 r 3 r 4 C 4 (c) Figure 4: Incremental drawing of M 4.

9 Crossing Number Bounds for the Mesh of Trees 9 edge crossed. Any drawing in which three 4-cycles have an edge crossed has at least four crossings, since, by an argument similar to that used in the proof of Lemma 4.1, at least one more crossing must occur involving the link edges between the cycles. case 1: Assume that two or more 4-cycles intersect. This adds at least two crossings. If more than two intersect then there are at least four crossings. Hence, at most two may intersect in at most two points otherwise, three or more crossings would occur and there would also be at least one other crossing involving a link edge of another 4-cycle. Now the other two 4-cycles may lie inside or outside the region R formed by the intersection of cycles C 1 and C 2. If one or both lie inside R then at least two of their link edges must cross either the boundary of R or other link edges between them. If both lie outside R but inside C 1 (or C 2 ), or one lies inside C 1 and the other lies inside C 2, then at least two of their link edges must cross either another 4-cycle or other link edges or else a single link edge must cross two other edges. If one or both lie outside C 1 and C 2 then, by a similar argument, at least two of their link edges must cross either another 4-cycle or other link edges, or else a single link edge must cross two other edges. Hence, there are at least four crossings. case 2: None of the 4-cycles intersect, and one 4-cycle, C 1, has an edge crossed. Then there are three subcases C 1 contains either one or two other 4-cycles and one or two other cycles lie outside C 1, or else C 1 either self-intersects or intersects one of its own edges. subcase 1: One 4-cycle C 2 lies inside C 1, and two other 4-cycles C 3 and C 4 lie outside of C 1. Two non-self-intersecting cycles are disjoint if the intersection of their interior regions is empty. Then C 3 and C 4 must be disjoint otherwise, there would be more than one 4-cycle with an edge crossed. Now C 1 must have at least two edge crossings since there are two link edges between C 2 and C 3. Also, since C 3 and C 4 are disjoint, at least two of their link edges must cross other link edges either between themselves or to C 2 and C 1, resulting in at least four crossings in all. subcase 2: Two 4-cycles C 2 and C 3 lie inside C 1, and the other 4-cycle C 4 lies outside of C 1. C 2 and C 3 must be disjoint otherwise, there would be more than one cycle with an edge crossed. The link edges from C 1 to C 2 divide the interior of C 1 into two regions, and C 3 must lie entirely within one of them. Then one of the link edges from C 2 to C 3 must cross a link edge

10 Crossing Number Bounds for the Mesh of Trees 10 from C 2 to C 1. Also, the two link edges from C 3 to C 4 must both cross the enclosing cycle C 1, and one link edge from C 3 to C 4 must cross a link edge from C 4 to C 1, yielding at least four crossings in all. subcase 3: If C 1 self-intersects then it is not a facial 4-cycle and the relative positions of two of its adjacent vertices, u and v, are interchanged with their relative positions in a facial 4-cycle. Since u and v are joined by link edges to two other 4-cycles, these link edges must still incur at least two crossings. If the other three facial 4-cycles are disjoint then they must generate at least one other crossing involving their link edges, yielding at least four crossings in all. If C 1 does not self-intersect then one of its link edges must intersect it. But then this link edge must incur at least one more crossing in joining with another 4-cycle, and the other 4-cycles must incur at least two more crossings involving their link edges. case 3: None of the 4-cycles intersect, and two 4-cycles C 1 and C 2 have an edge crossed. Then C 1 and C 2 must each contain another 4-cycle, C 4 and C 3, respectively, and the two link edges from C 3 to C 4 each cross the two enclosing cycles C 1 and C 2, yielding at least four crossings. Therefore, since any drawing of M 4 must satisfy one of the preceding two lemmas, the following theorem is immediate. Theorem 4.3. ν(m 4 ) = 4. 5 Lower Bounds for ν(m n ) In [11], p. 58, the following lower bound is given for ν(m n ): ν(m n ) (n 2 log n 121n n)/40 for n 1. However, this function is positive only for extremely large n, i.e., n 2 121, and hence is not useful for practical instances of M n. We derive a new lower bound which is positive for n 4. To obtain the bound, we use the recursive definition of M n. Recall that M n contains four copies of M n/2 as subgraphs. Disregarding link edges, the number of crossings must be at least the sum of the crossings in each of the

11 Crossing Number Bounds for the Mesh of Trees 11 four copies of M n/2 in M n. Hence, we have the inequality ν(m n ) 4ν(M n/2 ) together with the initial condition ν(m 4 ) = 4. Solving this recurrence, we obtain the following lower bound: Theorem 5.1. ν(m n ) n 2 /4 for n 4. We can improve upon this bound using the edge tracing approach of Leighton [11] and optimizing all steps of his method. Theorem 5.2. ν(m n ) (5n 2 log n 44n 2 )/80 for n 4. Proof: First note that ν(m n ) n 2 /4 by Theorem 5.1. Hence the theorem holds for n 256. Assume n 512. Let 2K n 2 denote the complete multigraph on n 2 vertices obtained from K n 2 by replacing each edge by two parallel edges. Let D be any drawing of M n in the plane. From this drawing, we can construct a drawing D of 2K n 2 in the following way. Locate the n 2 leaves of the binary trees of D, which will serve as the vertices for 2K n 2. Let the binary trees be referenced by their row 1,..., n from top to bottom in a drawing D such as that of Figure 1 where there are four row trees rooted along the y-axis. Given any pair (i, j) and (k, l) of these vertices, draw the first edge from (i, j) to (k, l) along the unique path from (i, j) to (i, l) in the i th row tree of D and then from (i, l) to (k, l) in the l th column of D. Similarly draw the second edge from (k, l) to (i, j) through (k, j). We next count the number of crossings in D. There are two types of crossings: the first kind results from a crossing in D involving edges of 2K n 2 occurring where two edges of M n cross, and the second kind results from edges of 2K n 2 which must cross while traversing a common edge of D. For crossings of the second kind, we need to count the number of times each edge of D is traced over during the construction of D. We organize the edges of a binary tree into levels according to their distance from the root. Hence, all edges incident with the root are in level 1, all edges incident with these edges are in level 2, etc. It is not difficult to show that each edge in the i th level of any row tree of M n (henceforth, referred to as a type i edge) is traced over at most 2 i+1 (1 2 i )n 3 times for any i log n during the construction of D. Thus at most 2 2i+1 (1 2 i ) 2 n 6 crossings of the second kind can occur at any type i edge of D. Since there are 2 i+1 n type i edges in M n, we can conclude that the number of crossings of the second kind, cr 2, in D is as

12 Crossing Number Bounds for the Mesh of Trees 12 follows: log n cr 2 2 i+1 n2 2i+1 (1 2 i ) 2 n n7. We next count the number of crossings of the first kind. We say that a crossing of D is type i-j if it is the crossing of a type i edge and a type j edge. Let t ij denote the number of type i-j crossings in D and let log n t i = t ij. j=i Since each type i edge is traced over at most 2 i+1 (1 2 i )n 3 times, each type i-j crossing of D produces at most 2 i j+2 (1 2 i )(1 2 j )n 6 crossings of the first kind in D. Thus the number of crossings of the first kind, cr 1, in D is as follows: log n log n log n cr 1 n 6 2 i j+2 (1 2 i )(1 2 j )t ij n 6 2 2i+2 (1 2 i ) 2 t i. j=i Summing, we find that the total number of crossings of either kind, cr, in D is as follows: cr = cr 1 + cr 2 40 log n 21 n7 + n 6 2 2i+2 (1 2 i ) 2 t i. Note that cr must be at least ν(2k n 2). Now recall from [22] that for any K n, n even, the following holds ν(k n ) n(n 1)(n 2)(n 3)(n 8). 80(n 7) Combining this with ν(2k n 2) = 4ν(K n 2) from [8], we obtain, after some calculations, Hence ν(2k n 2) 1 20 (n8 7n 6 ). log n 2 2i+2 (1 2 i ) 2 t i 1 20 (n2 40n).

13 Crossing Number Bounds for the Mesh of Trees 13 Let s k = k t i be the number of crossings involving at least one edge from the top k levels of some binary tree of M n. We will use the preceding inequality to show that s k 33(n 2 40n)k/512 for at least one value of k 1. Assume otherwise and observe that log n 2 2i+2 (1 2 i ) 2 t i = = log n 2 2i+2 (1 2 i ) 2 (s i s i 1 ) log n 1 (2 2i+2 (1 2 i ) 2 2 2i (1 2 i 1 ))s i < + 2 2log n+2 (1 2 log n )s log n log n 2 2i+2 (1 2 i ) 2 s i < 33 log n 512 (n2 40n) 2 2i+2 (1 2 i ) 2 i (1) < 33 log n 512 (n2 40n) 2 2i+2 (1 2 i ) (2) < 1 20 (n2 40n), a contradiction. Thus for all n 512, there is a k 1 such that s k 33(n 2 40n)k/512. By itself, the summation term in (1) is less than the summation term in (2) for n 4. However, the complete expression (1) is less than the complete expression (2) for n 64. The last inequality follows from the fact that the summation term in (2) evaluates to as n, and hence the entire expression evaluates to Finally, we show by induction on n that ν(m n ) (5n 2 log n 44n 2 )/80. The claim holds for n 256. Assume that the lower bound holds for all powers of 2 less than n and n 512. By counting the crossings of D in

14 Crossing Number Bounds for the Mesh of Trees 14 two groups according to whether or not at least one edge of the crossing is contained in the top k levels of the binary trees of M n, we obtain ν(m n ) 2 2k ν(m 2 n) + s ( k k 1 2 2k k n 2 (log n k) 11 ) k n ( n 2 40n ) k 512 = 1 16 n2 log n 11 ( 33n 20 n2 + kn 512 n ) n2 log n n2 = 5n2 log n 44n Remark. The induction in the end of the proof can be easily done in a more general way showing that ( ) (( ) ( ) ν(m n ) 512 ε n 2 log n 512 ε log 1 1 ) n 2, 8ε 4 for ε > 0. To simplify the expression for the lower bound we have chosen ε = 33/512 1/16 = In Table 1 we list the values of the three lower bounds for values of n up to 65, 536. The bound of Theorem 5.2 becomes tighter than the bound of Theorem 5.1 at n = The bound given in [11] is negative for n < An Upper Bound for ν(m n ) In this section we derive an upper bound for ν(m n ). We do this by defining, recursively, a suppressed 4-facial drawing D of M n. The four copies of M n/2 in M n are drawn in the four quadrants of the plane, one copy per quadrant, with each copy drawn 4-facially, and are joined by link edges between the roots of the row and column trees of each M n/2 subgraph. The drawing for M 8 is illustrated in Figure 5. The following lemma is used in computing an upper bound for ν(m n ) from D:

15 Crossing Number Bounds for the Mesh of Trees 15 Lemma 6.1. There are 2n link edges joining the four M n/2 subgraphs in M n. Proof: We first suppress all degree-2 vertices of M n, as before. Figure 5 illustrates this for M 8 after suppression of degree-2 vertices and contraction. There are now n/2 link edges connecting each M n/2 subgraph of one quadrant with its neighboring M n/2 subgraphs in the two adjacent quadrants. Since there are four quadrants, the lemma follows. We classify the link edges into two types, horizontal and vertical, of which there are n each in number. Lemma 6.2. Of the n horizontal (vertical) link edges, n 2 are involved in crossings, yielding a total of (n 4) crossings. Proof: We label the link edges 1,..., n from top to bottom. Edges 1 and n can be drawn without crossings, as shown in Figure 5 for M 8. Let the four quadrants of Figure 5 be labelled 1-4, starting with the upper left quadrant and proceeding clockwise. The n 4 horizontal link edges 2,..., n/2 1 and n/2+2,..., n 1, joining vertices of quadrant 1 (3) with quadrant 2 (4), each cross with n/2 2 edges of M n/2, adding a total of 2(n 4) crossings, as is shown in the figure. Similarly, the n 4 vertical link edges 2,..., n/2 1 and n/2 + 2,..., n 1 each cause n/2 2 crossings. Finally, the innermost pairs of horizontal and vertical link edges, edges n/2 and n/2 + 1, respectively, pairwise intersect, adding four crossings to the total. Adding up all crossings involving the link edges, we obtain a total of 2(n 4)(n/2 2)+4 = (n 4) 2 +4 crossings. The remaining crossings occur within the M n/2 subgraphs. This leads to the following recurrence for the number of crossings, cr D (M n ), in D: cr D (M n ) = 4cr D (M n/2 ) + (n 4) It is clear that any M n can be drawn as in D with the specified number of crossings. The initial conditions are cr D (M 2 ) = 0, cr D (M 4 ) = 4, and cr D (M 8 ) = 36. Solving the recurrence by standard techniques, we obtain the general solution c 1 n 2 + c 2 n 2 log n + c 3 n. Using the initial conditions, we arrive at the following result:

16 Crossing Number Bounds for the Mesh of Trees 16 Table 1: Comparison of the Three Lower Bounds for ν(m n ). 5n 2 log n 44n 2 80 n 2 log n 121n n 40 n n 2 / Theorem 6.3. ν(m n ) (log n 10 3 )n2 + 8n Remarks We have been unable to find drawings of M n with fewer crossings than the ones given here. In fact, we conjecture that equality actually holds in Theorem 6.3. References [1] M. Anderson, R.B. Richter, and P. Rodney, The crossing number of C 6 C 6, Congressus Numerantium 118 (1996) [2] S.N. Bhatt and F.T. Leighton, A framework for solving VLSI graph layout problems, J. Comput. & Sys. Sci. 28 (1984)

17 Crossing Number Bounds for the Mesh of Trees 17 Figure 5: A 4-facial suppressed drawing of M 8 with 36 crossings. [3] A.M. Dean and R.B. Richter, The crossing number of C 4 C 4, J. Graph Theory 19 (1) (1995) [4] P. Erdos and R.P. Guy, Crossing number problems, Amer. Math. Monthly 80 (1973) [5] L. Faria and C.M.H. de Figueiredo. On Eggleton and Guy s conjectured upper bound for the crossing number of the n-cube. Math. Slovaca 50 (3) (2000) [6] M.R. Garey and D.S. Johnson, Crossing number is NP-complete, SIAM J. Alg. Disc. Meth. 4 (3) (1983) [7] F. Harary and W.T. Tutte, A dual form of Kuratowski s theorem, Canad. Math. Bull. 8 (1965) [8] P.C. Kainen, A lower bound for crossing numbers of graphs with applications to K n, K p,q, and Q(d), J. Comb. Theory B 12 (1972)

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19 Crossing Number Bounds for the Mesh of Trees 19 [21] K. Wagner, Über eine eigenschaft der ebenen komplexe, Math. Ann. 114 (1937) [22] A.T. White and L.W. Beineke, Topological graph theory, in: L.W. Beineke and R.J. Wilson, eds., Selected Topics in Graph Theory, (Academic Press, 1978)