Calculating Crossing Numbers of Graphs Using Their Redrawings
Abstract
:1. Introduction
2. Cyclic Permutations and Possible Drawings of
- Let us first suppose that ; that is, . The number of crossings in D satisfies
- Now, let , which yields that and also that . By fixing the subgraph for some , we have
3. The Crossing Number of
- (a)
- Let be the nonempty set; that is, there is a subgraph . The reader can easily see that the subgraph is uniquely represented by . By fixing the subgraph , if edges of are crossed by any other subgraph at least five times, we obtainIf there is some subgraph with , then the vertex cannot be placed in the outer region of subdrawing with all six vertices of on its boundary, and enforces . Thus, by fixing the subgraph , we have
- (b)
- Let be the empty set; that is, there is a subgraph . As , we deal with possible configurations from the nonempty set . For any , if there is a subgraph , such that and with , the same fixation of like in the previous case also confirms a contradiction with (3) in D.Now, let us turn to the possibility of obtaining the minimum value 4 in Table 1; that is, could be achieved in D for two different . In the rest of the paper, assume that there are two different subgraphs such that and have mentioned configurations and , respectively. Then, holds for any with by summing two corresponding values in Table 1. We can easily verify in six possible regions of and that and are fulfilling for any , which yields that trivially holds for any such subgraph . Moreover, each of subgraphs of crosses at least six times. As , by fixing the subgraph , we haveThe obtained number of crossings contradicts the assumption (3). Finally, let us consider that holds for all with . By fixing the subgraph for some , we haveThis again confirms a contradiction with (3) in D.
4. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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6 | 4 | 4 | 5 | |
4 | 6 | 5 | 4 | |
4 | 5 | 6 | 4 | |
5 | 4 | 4 | 6 |
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Staš, M. Calculating Crossing Numbers of Graphs Using Their Redrawings. Symmetry 2023, 15, 175. https://doi.org/10.3390/sym15010175
Staš M. Calculating Crossing Numbers of Graphs Using Their Redrawings. Symmetry. 2023; 15(1):175. https://doi.org/10.3390/sym15010175
Chicago/Turabian StyleStaš, Michal. 2023. "Calculating Crossing Numbers of Graphs Using Their Redrawings" Symmetry 15, no. 1: 175. https://doi.org/10.3390/sym15010175