Abstract
The degree diameter problem explores the biggest graph (in terms of number of nodes) subject to some restrictions on the valency and the diameter of the graph. The restriction on the valency of the graph does not impose any condition on the number of edges (apart from taking the graph simple), so the resulting graph may be thought of as being embedded in the complete graph. In a generality of the said problem, the graph is taken to be embedded in any connected host graph. In this article, host graph is considered as the enhanced mesh network constructed from the grid network. This article provides some exact values for the said problem and also gives some bounds for the optimal graphs.
1. Introduction
All graphs discussed in paper are simple, finite, and undirected. The valency of a node in the graph is the number of edges connected with that node in . The maximal valency of the graph is indicated by . The distance between two nodes and of is the length of the shortest path between them. The distance between a node and the set is defined as . For any , let denote the subgraph of after removing from all the nodes of and all the edges incident to at least one node of . The diameter of the graph is designated by and is described as the largest distance between any pair of nodes of the graph . For a given graph and natural numbers and , indicates the number of nodes of largest subgraph of with given maximal valency and given diameter .
The topology of a network such as a telecommunications, multiprocessor, or local area network is generally represented by a graph such that stations or processors are represented by vertices (or nodes) and the links or the connection between these networks are represented by edges. There are many important features in the designing of such networks. One of the important aspects is to put limitation on vertex degree and its diameter. These two parameters in networks are interpreted as follows. By the degree of a node, it is meant to have the number of connections attached to that node; on the contrary, the diameter shows the largest number of links that must be required to transmit a message between any two nodes. The natural question that arises in this case is
“What is then the largest number of nodes in a network with a limited degree and diameter?” If we design the network so that there is no directed edge, then this leads to the Degree/Diameter Problem. More formally, we define it as follows.
Find the largest possible number of vertices in a graph of maximum degree and diameter .
For a thorough survey of the state of the problem, see [1, 2]. In the Degree Diameter Problem, only restriction that is imposed on the edges is the maximum degree, so there is considerable freedom in placing edges so as to avoid violating the diameter constraint. In this way, the resulting graph may be thought of as being embedded in the complete graph. In this case, the complete graph is acting as the host graph.
A generalization of the Degree Diameter Problem is to consider the graph as embedded in some host graph, not necessarily the complete graph. This problem becomes more interesting when the host graph is considered as a graph obtained from some network. The problem was first posed by Dekker et al.[7] in the following form.
Given connected undirected host graph , an upper bound for the maximum degree, and an upper bound for the diameter find the largest connected subgraph of maximum degree and diameter .
In [3–6], the degree diameter problem of honeycomb network, triangular network, oxide network, and silicate network has been explored. In [7], the largest subgraph has been determined with multidimensional hexagonal grid as the host graph. In [8–15], some extremal properties of graph networks are discussed. Mesh networks can relay messages using either a flooding technique or a routing technique. With routing, the message is propagated along a path by hopping from node to node until it reaches its destination. To ensure that all its paths are available, the network must allow for continuous connections and must reconfigure itself around broken paths, using self-healing algorithms such as Shortest Path Bridging. Self-healing allows a routing-based network to operate when a node breaks down or when a connection becomes unreliable. As a result, the network is typically quite reliable as there is often more than one path between a source and a destination in the network. Although mostly used in wireless situations, this concept can also apply to wired networks and to software interaction. In this work, we have extended this study to the enhanced mesh network.
The Cartesian product of the graphs and is the graph with node set , where two nodes and are adjacent if and only if either and or and .
Let and are two paths having nodes and , respectively. The graph of the grid network is obtained by their Cartesian product. The grid graph network has nodes and edges. The graph of the two-dimensional infinite grid network is denoted by .
Let be the graph of the enhanced mesh obtained from by replacing its each by a wheel , the hub of the wheel being a new vertex (see Figure 1). In the graph , each 4-cycle of is divided into four triangles where every node lie on some triangle.
(a)
(b)
Let us define the edges of the wheel as follows:(i)The edge connected to the hub and left-end vertex of upper horizontal edge in is called upper-left hub edge(ii)The edge connected to the hub and right-end vertex of upper horizontal edge in is called upper-right hub edge(iii)The edge connected to the hub and left-end vertex of lower horizontal edge in is called lower-left hub edge(iv)The edge connected to the hub and right-end vertex of lower horizontal edge in is called lower-right hub edge
Since has squares (4-cycles). Therefore, has nodes. In , the edges adjacent to the hub vertex in each are disjoint; therefore, has edges. Furthermore, we define that any two wheel graphs in are said to be adjacent if they share an edge.
2. Methodology
In this paper, we have calculated in the infinite enhanced mesh network. First, we find the induced closed balls of diameter for whose order is for . The order of is the upper bound for . We find for by deleting minimum number of vertices of . The rest of the paper is ordered as follows. In Section 3, we consider the case for ; in Section 4, the case for is discussed.
3. Result and Discussion
In this section, the results obtained are discussed. The largest subgraphs are obtained for the enhanced mesh networks for given degree and diameter.
3.1. Largest Subgraphs for
Let denote the infinite Enhanced Mesh Network in the Cartesian plane with .
Proposition 1. For , let the induced subgraph of of diameter and maximal degree 8 is denoted by . Then, for ,
Proof. For even diameter , let be a closed ball having radius with center as an eight degree node of with node set . To find the nodes of , we draw horizontal lines on the nodes of and count the nodes on these horizontal lines by adding them from top to bottom.
Thus, for , , we haveNow, for odd diameter , let be a closed ball having radius and center as the fixed triangle with node set .
Then, by counting the nodes on horizontal lines, we have for , ,In Figure 2, the graphs for , and 11 are depicted. The central vertex of for even is depicted by and for odd diameter , the nodes of central triangle of are indicated by and is depicted by . It is important to note that the distance of any node from is defined as , where the node is the nearest node of from .
Let be the biggest connected subgraph of with maximal degree .
Then, for , is the triangle ; thus, .
For , the biggest subgraph of with is the closed ball itself. Thus, , for .
Hence, we showed the following statement.
(a)
(b)
(c)
(d)
Theorem 1. Let be the graph of enhanced mesh and be a positive integer. Then,Note that the values in the abovementioned theorem are also trivial upper bounds on for . Thus, we prove the following corollary.
Corollary 1. Let be the biggest connected subgraph of of maximal degree 8. Let be positive integers with . Then,
4. Largest Subgraphs for
We are interested to find the biggest subgraph of with given maximal valency and diameter . Since , it makes perception to consider the cases for . For , . Now, we consider the other cases.
4.1. Values for
If , then the biggest connected subgraph of is just an edge and hence .
Now, we discuss the case when .
Theorem 2. Let be the enhanced mesh network, and let be a natural number. Then,
Proof. For . Let and be the horizontal and vertical lines, respectively, that are passing through the central node of and divide the graph into four regions (upper left, upper right, lower left, and lower right), as shown in Figure 3. Consider a border cycle of the lower-left region containing the central node (see the blue cycle in Figure 3(a)). Cycle has nodes.
For . Let be the horizontal line passing through the edge and be the vertical line passing through the node of the triangle in the graph that divide the graph into four regions (upper left, upper right, lower left, and lower right), as shown in Figure 3. Consider a border cycle of the lower-left region containing the edge (see the blue cycle in Figure 3(b)). The cycle has nodes.
(a)
(b)
4.2. Values for
Theorem 3. Let be an even natural number, and let be an infinite enhanced mesh network. Then,
Proof. For , the graph contains , which is the biggest subgraph of of valency 7, since in the graph (shown in Figure 3(a)) of maximal valency , the central node can be attached to at most 7 nodes. Hence, .
For , and , we construct a subgraph (shown in Figure 4) from as follows.
Let be the central horizontal line passing through the central node , be the upper neighboring row, and be the lower neighboring row. Let be the wheel graph of lying on and above the central line at a distance zero on the right side of the central node . Furthermore, suppose that is the hub node of :(i)Delete the upper-left and upper-right hub edges in each lying on and above the line of (ii)Delete the lower-left and lower-right hub edges in each lying on and below the line of (iii)Delete the lower-left hub edge (lower-right hub edge) on left of the central node (on the right of the central node ) in each lying between the horizontal lines and of , except the wheel graph whose lower-left hub edge is deleted(iv)Delete the upper-left hub edge (upper-right hub edge) on left of the central node (on right of the central node ) in each lying between the lines and of the graph The resulting subgraph is denoted by (shown in Figure 4, for ) which is the spanning subgraph of ; hence, . Furthermore, graph also has diameter since the distance of all the nodes of from is the same as in except the node . However, , for all . This implies that is the biggest subgraph of of maximal valency and diameter .
Theorem 4. Let be an odd natural number, and let be the infinite enhanced mesh network. Then,
Proof. For , the graph contains and this is a biggest induced subgraph of of maximal valency 7. Hence, .
For , and, we construct a subgraph (shown in Figure 5) from as follows.
Let be the horizontal line passing through edge of the central triangle of graph and be the upper neighboring horizontal row. Let be the wheel subgraph of containing the central triangle :(i)Delete the upper-left and upper-right hub edges in each lying on and above the line of (ii)Delete the lower-left and lower-right hub edges in each lying on and below the line of (iii)Delete the upper-left, lower-left hub edges in each lying on the left of wheel graph .(iv)Delete the upper-right and lower-right hub edges in each lying on the right of wheel graph (v)Furthermore, delete the left and right vertical edges in the wheel graph The resulting subgraph is denoted by (shown in Figure 5 for ) is the spanning subgraph of ; hence, . Furthermore, the graph also has diameter since the distance of all the nodes of from the central triangle is the same as in . This implies that is the biggest subgraph of of maximal valency and diameter .
4.3. Values for
Theorem 5. Let be an even natural number, and let be the infinite enhanced mesh network. Then,
Proof. For , the graph contains . This is a biggest subgraph of of maximal valency 6 since in the graph (shown in Figure 3(a)) of maximal valency , and the central node can be connected to at most 6 nodes. Hence, .
For , , we construct a subgraph (shown in Figure 6) from as follows.
Let and be the horizontal and vertical lines passing through the central node that divide into four regions (upper left, upper right, lower left, and lower right). Let and be the wheel graphs of lying in the upper-right and lower-left regions at a distance 0 from the central node . Furthermore, suppose that and be the hub vertices of and , respectively:(i)Delete the upper-left, upper-right, and lower-left hub edges in each lying in the upper-left region of (ii)Delete the upper-left, upper-right, and lower-right hub edges in each lying in the upper-right region of except the wheel graph (iii)Delete the lower-left, lower-right, and upper-left hub edges in each lying in the lower-left region of except the wheel graph (iv)Delete the lower-left, lower-right, and upper-right hub edges in each lying in the lower-right region of (v)Furthermore, delete the upper-left, upper-right, lower-left hub edges, and right vertical edge in ; also, delete the lower-left, lower-right, upper-right hub edges, and left vertical edge in The resulting subgraph is denoted by (shown in Figure 6, for ), which is the spanning subgraph of ; hence, . Furthermore, the graph also has diameter since the distance of all the nodes of from is the same as in except the nodes and . However, , for all . This implies that is the biggest subgraph of of maximal valency and diameter .
Theorem 6. Let be an odd natural number, and let be the infinite enhanced mesh network. Then,
Proof. For , the graph contains and this is a biggest induced subgraph of of maximal valency 6. Hence, .
For , the graph of maximal valency 8 is a closed ball of radius 3 (shown Figure 3(b)). For maximal valency , the biggest subgraph cannot contain all the nodes of the graph , otherwise . Thus, the graph shown in Figure 7(a) is the biggest subgraph of maximal valency . Hence, .
For , and , we construct a subgraph (shown in Figure 7) from as follows.
Let be the wheel graph that contains the central triangle . Let be the horizontal row of the wheel graphs containing the wheel graph . Let be the wheel graphs lying on left (right) of and is adjacent to it. Also, let be the wheel graph below the wheel graph and is adjacent to it:(i)Delete the upper-left and upper-right hub edges in each lying above the row of (ii)Delete the lower-left and lower-right hub edges in each lying below the row of (iii)Delete the upper-left and lower-left hub edges in each lying on the row and left of wheel graph (iv)Delete the upper-right and lower-right hub edges in each lying on the line and right of wheel graph (v)Delete the vertical edges in each lying on the row (vi)Furthermore, delete the upper-right edge of , lower-left edge of , and upper-left edge of the The resulting subgraph is denoted by (shown in Figure 7, for ), which is the spanning subgraph of ; hence, . Furthermore, the graph also has diameter since the distance of all the nodes of from the central triangle is the same as in . This implies that is the biggest subgraph of of maximal valency and diameter .
4.4. Values for
Theorem 7. Let be an even natural number, and let be the infinite enhanced mesh network. Then,
Proof. For , the graph contains . This is the biggest subgraph of of maximal valency 5, since in graph (shown in Figure 3(a)) of maximal valency , the central node can be connected to at most 5 nodes. Hence, .
For , the graph shown in Figure 8 is of maximal valency 5 and diameter 4, which is the subgraph of (shown in Figure 3(a)). Hence, .
For , , we construct subgraph (shown in Figure 8) from as follows.
Let and be the horizontal and vertical lines passing through the central node that divide into four regions (upper left, upper right, lower left, and lower right). Let , , and be the wheel graphs of lying in the upper-left, upper-right, and lower-left regions, respectively, at a distance zero from the central node . Furthermore, suppose that , , and be the hub vertices of , , and , respectively:(i)Delete the upper-right and lower-left hub edges in each lying in the upper-left region of (ii)Delete the upper-right and lower-left hub edges in each lying in the lower-right region of (iii)Delete the upper-left and lower-right hub edges in each lying in the upper-right region of (iv)Delete the upper-left and lower-right hub edges in each lying in the lower-left region of (v)Delete all the horizontal edges in except the edges lying on the horizontal line (vi)Delete all the vertical edges in except the edges lying on the vertical line (vii)Delete the three oblique edges of , , and that are adjacent to the central node (viii)Now reinstate the left vertical edge of , right vertical edge of , and lower horizontal edge of The resulting subgraph is denoted by (shown in Figure 7, for ), which is the spanning subgraph of ; hence, . Furthermore, the graph also has diameter since the distance of all the nodes of from is the same as in except the nodes , , and . However, and , for all . This implies that is the biggest subgraph of of maximal valency and diameter .
Theorem 8. Let be an odd natural number, and let be the infinite enhanced mesh network. Then,
Proof. For , the graph contains and this is a biggest induced subgraph of of maximal valency 5. Hence, .
For , the graph shown in Figure 9(a) is of maximal valency and diameter 3, which is a subgraph of (shown in Figure 3(b)). Hence, .
For , the graph shown in Figure 9(a) is of maximal valency and diameter 5, which is a subgraph of (shown in Figure 3(b)). Hence, .
For , and , we construct a subgraph (shown in Figure 9) from as follows.
Let be the horizontal line passing through the edge of the central triangle and be the vertical column of wheel graphs containing the central triangle that divide the graph into four regions (upper left, upper right, lower left, and lower right), as shown in Figure 9(b). Let be the wheel graph that contains the central triangle . Let and be the wheel graphs lying in the lower-left and lower-right regions, respectively, at a distance zero from the central triangle , both lying on the line . Let be the wheel graph lying below the wheel graph and is adjacent to it:(i)Delete the upper-right and lower-left hub edges in each lying in the upper-left region of (ii)Delete the upper-right and lower-left hub edges in each lying in the lower-right region of (iii)Delete the upper-left and lower-right hub edges in each lying in the upper-right region of (iv)Delete the upper-left and lower-right hub edges in each lying in the lower-left region of (v)Delete all the horizontal edges of in the four regions except the edges that are lying on the horizontal line (vi)Delete all the vertical edges of in the four regions except the edges that are lying on the vertical column (vii)Delete the left and right vertical edges of wheel graph (viii)Delete the upper-left and upper-right hub edges in the wheel graph (ix)Delete the upper-right (upper-left) hub edge of the wheel graph (x)Now, reinstate the left vertical edge of and right vertical edge of The resulting subgraph is denoted by (shown in Figure 9, for ), which is the spanning subgraph of ; hence, . Furthermore, the graph also has diameter since the distance of all the nodes of from the central triangle is the same as in except the nodes , , and whose distance from central triangle is . This implies that is the biggest subgraph of of maximal valency and diameter .
4.5. Values and Bounds for
Now, we discuss the case when .
For even , we prove the succeeding theorem.
Theorem 9. Let be the infinite enhanced mesh and be an even natural number. Then,
Proof. The lower bound for is 5, since the graph contains . The biggest subgraph of of maximal valency 8 is the graph (shown Figure 3(a)). When , the central node of can be connected to at most four vertices. Therefore, we have to remove at least four vertices, which are adjacent to , otherwise . This implies that the biggest subgraph of of maximal valency 4 can contain at most 5 vertices. Hence, .
For , the graph contains the subgraph (shown in Figure 10(a)) that has maximal valency and diameter 4. Thus, .
For , the graph contain the subgraph (shown in Figure 10(a)) that has maximal valency 4 and diameter 6. Thus, .
For , and , we construct a subgraph (shown in Figure 10) from as follows.
Let and be the horizontal and vertical lines, respectively, passing through the central node that divide into four regions (upper left, upper right, lower left, and lower right). Let , , , and be the wheel graphs of lying in the upper-left, upper-right, lower-left, and lower-right regions, respectively, at a distance 1 from the central node and lying on the line . Let be the path of length passing through the central node and lies in upper-left and lower-right regions. Let be the path of length passing through the central node and lies in upper-right and lower-left regions. Furthermore, suppose that the nodes and lie on the path and and lie on the path that are adjacent to the central node :(i)Delete the upper-right and lower-left hub edges in each lying in the upper-left region of (ii)Delete the upper-right and lower-left hub edges in each lying in the lower-right region of (iii)Delete the upper-left and lower-right hub edges in each lying in the upper-right region of (iv)Delete the upper-left and lower-right hub edges in each lying in the lower-left region of (v)Delete all the horizontal edges in except the edges lying on the line (vi)Delete all the vertical edges in except the edges lying on the line (vii)Delete the four oblique edges , , , and that are adjacent to the central node (viii)Delete both the end vertices of paths and (ix)Now, reinstate the upper-right hub edge of , upper-left hub edge of , lower-right hub edge of , and lower-left hub edge of The resulting subgraph is denoted by (shown in Figure 10 for ). Furthermore, the graph has also diameter since the distance of all the nodes of from is the same as in except the nodes , , , and whose distance from is for all . Since is obtained from by deleting only four nodes of , which implies that .
Hence, .
Theorem 10. Let be an odd natural number, and let be the infinite enhanced mesh network. Then,
Proof. For , the graph contains and this is a biggest induced subgraph of of maximal valency 4. Hence, .
For , the graph shown in Figure 11(a) is of maximal valency and diameter 4, which is a subgraph of (shown in Figure 3(b)). Hence, .
For , the graph shown in Figure 11(a) is of maximal valency and diameter 5, which is a subgraph of (shown in Figure 3(b)). Hence, .
For , the graph shown in Figure 11(b) is of maximal valency and diameter 7, which is a subgraph of (shown in Figure 3(b)). Hence, .
For , and , we construct a subgraph (shown in Figure 11) from as follows
Let be the horizontal line passing through the edge of the central triangle and be the vertical column of wheel graphs containing the central triangle that divide the graph into four regions (upper left, upper right, lower left, and lower right). Let be the wheel graph that contains the central triangle . Let and be the wheel graphs lying in the lower-left and lower-right regions, respectively, at a distance 1 from the central triangle , both lying on the line . Let be the wheel graph lying below the wheel graph and is adjacent to it. Let be the wheel graph lying on left (right) of the wheel graph and is adjacent to it. Let be the wheel graph lying above the wheel graph and is adjacent to it:(i)Delete upper-right and lower-left hub edges in each lying in the upper-left region of (ii)Delete the upper-right and lower-left hub edges in each lying in the lower-right region of (iii)Delete the upper-left and lower-right hub edges in each lying in the upper-right region of (iv)Delete the upper-left and lower-right hub edges in each lying in the lower-left region of (v)Delete all the horizontal edges of in the four regions except the edges that are lying on the line (vi)Delete all the vertical edges of in the four regions except the edges that are lying in the column (vii)Delete the left and right vertical edges of (viii)Delete the upper-left, upper-right, and lower-right hub edges in each lying in the column and above except (ix)Delete the lower-left, lower-right, and upper-left hub edges in each lying in the column and below except (x)Delete the upper-left, upper-right, and lower-right hub edges in (xi)Delete the lower-left, lower-right, and upper-left hub edges in (xii)Furthermore, delete the edges , , , and (xiii)Now, reinstate the lower horizontal edges in and (xiv)Now, reinstate the upper horizontal edges in and The resulting subgraph is denoted by (shown in Figure 11(b), for ), which is the spanning subgraph of ; hence, . Furthermore, the graph also has diameter since the distance of all the nodes of from the central triangle is the same as in except the nodes , , , , , , , and whose distance from central triangle is . This implies that is the biggest subgraph of of maximal valency and diameter .
4.6. Bounds for
For , the graph contains and this is a biggest induced subgraph of of maximal valency 3. Hence, .
The lower bound for is 4, since the graph contains . The biggest subgraph of of maximal valency 8 is the graph (shown in Figure 3(a)). When , the central node of can be connected to at most three nodes. Therefore, we have to remove at least three nodes which are adjacent to , otherwise . This implies that the biggest subgraph of of maximal valency 3 can contain at most 4 nodes. Hence, .
Theorem 11. Let be the infinite enhanced mesh network and be an even natural number. Then,
Proof. For , the subgraphs of shown in Figure 12 are of maximal valency . It is also easy to check that the diameter of each graph is . Hence, the theorem is true for .
For , is the biggest subgraph of of maximal valency and diameter . Let be the subgraph of of maximal valency and diameter . The structure of the graph for is depicted in Figure 12. The central node of graph correspond to the central node of and is exhibited by . The whole graph is constructed from by deleting those nodes which have distance greater than from the central node . The structure of the deleted nodes in the construction of is shown in Figure 12. To calculate the deleted nodes of , we draw horizontal lines on deleted nodes and counting them in horizontal rows from top to bottom:
Delete vertices in .
Hence, for .
Furthermore, for every node . Thus, has diameter at most .
Hence, the theorem is satisfied for .
Theorem 12. Let be the infinite enhanced mesh network and be an odd natural number. Then,
Proof. For , the subgraphs of shown in Figure 13 are of maximal valency . It is also easy to check that each graph has a diameter . Hence, the theorem is true for .
For , is the biggest subgraph of of maximal valency and diameter . Let be the subgraph of of maximal valency and diameter . The structure of the graph for is exhibited in Figure 13. The central triangle of correspond to the central triangle of and is depicted by . The whole graph is constructed from by deleting those nodes which have distance greater than from the central triangle . The structure of the deleted nodes in the construction of is shown in Figure 13. Now, calculate the deleted nodes of in horizontal rows from top to bottom.
Delete Vertices in .
Hence, for , .
Furthermore, for every node . Thus, has diameter at most .
Hence, the theorem is satisfied for .
Using the result from Theorems 11 and 12 and the fact that for any graph, we get the succeeding statement.
Corollary 2. Let be the infinite enhanced mesh network and let is a natural number. Then, for ,
5. Conclusion
In the Degree Diameter Problem, the largest graphs in term of vertices are computed with given degree and diameter. In this work, we have considered the restricted version of Degree Diameter Problem which states that given connected undirected host graph , an upper bound for the maximum degree, and an upper bound for the diameter find the largest connected subgraph of maximum degree and diameter . This problem becomes of particular interest when we consider the host graph as the network.
The topology of a network (such as a telecommunications, multiprocessor, or local area network, to name just a few) is usually modeled by a graph in which vertices represent ‘nodes’ (stations or processors) while undirected or directed edges stand for ‘links’ or other types of connections. There are many important features in the designing of such networks. One of the important aspects is to put limitation on vertex degree and its diameter. These two parameters in networks are interpreted as follows. By the degree of a node, it is meant to have the number of connections attached to that node; on the contrary, the diameter shows the largest number of links that must be required to transmit a message between any two nodes. The natural question that arises in this case is “what is then the largest number of nodes in a network with a limited degree and diameter?” If we design the network so that there is no directed edge, then this leads to the Degree/Diameter Problem.
Planar graphs are popular for network design because they have a convenient physical layout. Of all planar network topologies, the (square) mesh is the most popular and this has been studied in [1, 3–5, 7]. In this work, we have extended this study by considering the host graph as the graph obtained from the enhanced mesh network.
In future, it will be interesting to find more such mesh networks and study the optimal subgraphs.
Furthermore, we can extend this study to directed networks.
Data Availability
Data from previous studies were used to support this study. They are cited at relevant places within the text as references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by UPAR Grants of United Arab Emirates University (UAEU) under Grant nos. G00002590 and G00003271.