Edge Deletions

Linked lists—Data items linked by pointers. In the general form, each item, except the first, has one predecessor, and each item, except the last, has one successor, with pointers linking items to their successors. Doubly linked lists have pointers to both the predecessor and the successor of an item and a circular list has a pointer from the final item to the initial item (producing a predecessor to the initial item and a successor to the final item). Restricted lists also exist, such as stacks, where items may only be added (pushed) or deleted (popped) at one end (the top), and queues, where items must be inserted at one end and deleted from the other. Trees are linked lists in which each item (node) except the root node has one predecessor, but all nodes may have any finite number, or zero, successors graphs contain both nodes and edges, which connect the nodes and define their relationships. 

The second cell-specific regulatory element within the distal enhancer is named SER. Loss of this element, which is at the extreme distal edge of the enhancer, leads to a selective loss of DDC expression in the ventral lateral serotonin cells (Johnson et al., 1989 Lundell and Hirsh, 1992) (Fig. 6C). This element has been delimited to about 40 bp and shows unexpected complexity, consisting of two functionally redundant elements. These two subelements, SERl and SERr, are each sufficient to allow normal DDC expression in the ventral lateral serotonin neurons if the other is deleted. In spite of this functional similarity, no sequence similarity is apparent between the two regions. The region of conservation between D. melanogaster and D. virilis is limited to SERl. 

We shall now offer two simple examples to illustrate these points. Example 1 shown in Fig. 8a consists of 2 independent cycles. Starting from vertex 1, we delete edges 1 and 2 to locate (1, 2, 3, 4, 5) as cycle 1. Similarly, starting from either vertex 2 or vertex 5, the algorithm traces (3,4,5,6,7,8,9,... 

Example 2 shown in Fig. 8b contains 4 independent cycles. Again starting from vertex 1 and deleting edges 1 and 2, we obtain (1, 2, 4, 7) as cycle 1. Similarly, with vertices 2 and 3 as key nodes the Epp-Fowler algorithm yields (4, 5, 8) and (6, 7,8) as the two remaining cycles. In this case it would appear that only 3 of the 4 independent cycles are located by the algorithm. 

Sachs algorithm can be used on coronoid systems, but the deletion method of convex pair can not. An example is shown in Fig. 7. (u, v) is a convex pair in the coronoid system, but in any Kekule structure of the system, the edge uv is not a double bond edge. 

A generalized benzenoid system is obtained by deleting some vertices and edges from a benzenoid system [14]. A generalized benzenoid system may be disconnected then each independent conjugated subsystem is called a component. A benzenoid system is a special case of a generalized benzenoid system. 

Suppose H has an equal number of peaks and valleys. We shall select some vertices and edges, such that deleting them does not influence the existence of Kekule structures. There are two cases. 

The algorithm comes to a stop when either (a) a non-Kekulean condition is reached then H is non-Kekulean, or (b) all vertices of H are deleted in this case H is Kekulean. If H is recognized as Kekulean, we recall the deleting process if we delete two end vertices of an edge, then the edge is considered as a double bond, and in this way we can constitute a Kekule structure of H. 

The above procedure ends when an isolated vertex is created or when all vertices of H are pairwise deleted. In the former case H is non-Kekulean in the latter case it is Kekulean, and a Kekule structure can be constructed by regarding the edge joining the deleted pair of vertices at each step as a double bond. 

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