# Turning Cliques into Paths to Achieve Planarity

• Patrizio Angelini
• Seok-Hee Hong
• Karsten Klein
• Stephen Kobourov
• Giuseppe Liotta
• Alfredo Navarra
• Alessandra Tappini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)

## Abstract

Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call $$h$$ -Clique2Path Planarity: Given a graph G, whose vertices are partitioned into subsets of size at most h, each inducing a clique, remove edges from each clique so that the subgraph induced by each subset is a path, in such a way that the resulting subgraph of G is planar. We study this problem when G is a simple topological graph, and establish its complexity in relation to k-planarity. We prove that $$h$$ -Clique2Path Planarity is NP-complete even when $$h=4$$ and G is a simple 3-plane graph, while it can be solved in linear time, for any h, when G is 1-plane.

## References

1. 1.
Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Rutter, I.: Intersection-link representations of graphs. J. Graph Algorithms Appl. 21(4), 731–755 (2017).
2. 2.
Angelini, P., Eades, P., Hong, S.-H., Klein, K., Kobourov, S., Liotta, G., Navarra, A., Tappini, A.: Turning cliques into paths to achieve planarity. CoRR 1808.08925v2 (2018). http://arxiv.org/abs/1808.08925v2
3. 3.
Bekos, M.A., Kaufmann, M., Raftopoulou, C.N.: On optimal 2- and 3-planargraphs. In: 33rd International Symposium on Computational Geometry, SoCG2017, 4-7 July 2017, Brisbane, Australia, pp. 16:1–16:16 (2017).
4. 4.
Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M.: Computing NodeTrix representations of clustered graphs. J. Graph Algorithms Appl. 22(2), 139–176 (2018).
5. 5.
Di Giacomo, E., Liotta, G., Patrignani, M., Tappini, A.: NodeTrix planarity testing with small clusters. In: Frati, F., Ma, K.-L. (eds.) GD 2017. LNCS, vol. 10692, pp. 479–491. Springer, Cham (2018).
6. 6.
Didimo, W., Liotta, G., Montecchiani, F.: A survey on graph drawing beyond planarity. CoRR abs/1804.07257 (2018). http://arxiv.org/abs/1804.07257
7. 7.
Ghoniem, M., Fekete, J., Castagliola, P.: On the readability of graphs using node-link and matrix-based representations: a controlled experiment and statistical analysis. Inf. Vis. 4(2), 114–135 (2005).
8. 8.
Henry, N., Fekete, J., McGuffin, M.J.: NodeTrix: a hybrid visualization of social networks. IEEE Trans. Vis. Comput. Graph. 13(6), 1302–1309 (2007).
9. 9.
Kindermann, P., Klemz, B., Rutter, I., Schnider, P., Schulz, A.: The partition spanning forest problem. In: Mulzer, W. (ed.) Proceedings of the 34th European Workshop on Computational Geometry (EuroCG 2018), Berlin (2018, to appear)Google Scholar
10. 10.
Kobourov, S.G., Liotta, G., Montecchiani, F.: An annotated bibliography on 1-planarity. Comput. Sci. Rev. 25, 49–67 (2017).
11. 11.
Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM 55(2), 11:1–11:29 (2008).
12. 12.
Okoe, M., Jianu, R., Kobourov, S.: Revisited experimental comparison of node-link and matrix representations. In: Frati, F., Ma, K.-L. (eds.) GD 2017. LNCS, vol. 10692, pp. 287–302. Springer, Cham (2018).
13. 13.
Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997).
14. 14.
Yang, X., Shi, L., Daianu, M., Tong, H., Liu, Q., Thompson, P.: Blockwise human brain network visual comparison using NodeTrix representation. IEEE Trans. Vis. Comput. Graph. 23(1), 181–190 (2017).

© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Patrizio Angelini
• 1
• 2
• Seok-Hee Hong
• 2
• Karsten Klein
• 3
• Stephen Kobourov
• 4
• Giuseppe Liotta
• 5
• Alfredo Navarra
• 5
• Alessandra Tappini
• 5
Email author
1. 1.University of TübingenTübingenGermany
2. 2.The University of SydneySydneyAustralia
3. 3.University of KonstanzKonstanzGermany
4. 4.University of ArizonaTucsonUSA
5. 5.University of PerugiaPerugiaItaly