Abstract
Given a graph G and a pair \(\langle e',e''\rangle \) of distinct edges of G, an
is an operation that turns G into a new graph \(G'\) by subdividing edges \(e'\) and \(e''\) with a dummy vertex \(v'\) and \(v''\), respectively, and by adding the edge \((v',v'')\). In this paper, we show that any 2-connected simple planar graph with minimum degree \(\delta (G) \ge 3\) and maximum degree \(\varDelta (G)\) can be augmented by means of edge-edge additions to a 3-connected planar graph \(G'\) with \(\varDelta (G') = \varDelta (G)\), where each edge of G participates in at most one edge-edge addition. This result is based on decomposing the input graph into its 3-connected components via SPQR-trees and on showing the existence of a planar embedding in which edge pairs from a special set share a common face. Our proof is constructive and yields a linear-time algorithm to compute the augmented graph.
As a relevant application, we show how to exploit this augmentation technique to extend some classical NP-hardness results for bounded-degree 2-connected planar graphs to bounded-degree 3-connected planar graphs.
This research was supported in part by MIUR Project “MODE” under PRIN 20157EFM5C, by MIUR Project “AHeAD” under PRIN 20174LF3T8, by H2020-MSCA-RISE project 734922 – “CONNECT”, by MIUR-DAAD JMP N\(^\circ \) 34120, and by Ru 1903/3-1 of the German Science Foundation (DFG).
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References
Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Roselli, V.: Relaxing the constraints of clustered planarity. Comput. Geom. 48(2), 42–75 (2015)
Angelini, P., Da Lozzo, G., Neuwirth, D.: Advancements on SEFE and partitioned book embedding problems. Theor. Comput. Sci. 575, 71–89 (2015)
Barnette, D.W.: On Steinitz’s theorem concerning convex 3-polytopes and on some properties of planar graphs. In: Chartrand, G., Kapoor, S.F. (eds.) The Many Facets of Graph Theory. LNM, vol. 110, pp. 27–40. Springer, Heidelberg (1969). https://doi.org/10.1007/BFb0060102
Bertolazzi, P., Di Battista, G., Liotta, G., Mannino, C.: Upward drawings of triconnected digraphs. Algorithmica 12(6), 476–497 (1994)
Biedl, T.C., Kant, G., Kaufmann, M.: On triangulating planar graphs under the four-connectivity constraint. Algorithmica 19(4), 427–446 (1997)
Da Lozzo, G., Rutter, I.: Strengthening hardness results to 3-connected planar graphs. CoRR abs/1607.02346 (2016). http://arxiv.org/abs/1607.02346
Dailey, D.P.: Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete. Discret. Math. 30(3), 289–293 (1980)
Di Battista, G., Tamassia, R.: On-line graph algorithms with SPQR-trees. In: Paterson, M.S. (ed.) ICALP 1990. LNCS, vol. 443, pp. 598–611. Springer, Heidelberg (1990). https://doi.org/10.1007/BFb0032061
Didimo, W., Liotta, G., Patrignani, M.: Bend-minimum orthogonal drawings in quadratic time. In: Biedl, T., Kerren, A. (eds.) GD 2018. LNCS, vol. 11282, pp. 481–494. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-04414-5_34
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)
Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)
Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR-trees. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 77–90. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44541-2_8
Schmidt, J.M.: Construction sequences and certifying 3-connectedness. In: Marion, J., Schwentick, T. (eds.) STACS 2010. LIPIcs, vol. 5, pp. 633–644. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2010)
Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)
Whitney, H.: Congruent graphs and the connectivity of graphs. Am. J. Math. 54(1), 150–168 (1932)
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Da Lozzo, G., Rutter, I. (2019). Reaching 3-Connectivity via Edge-Edge Additions. In: Colbourn, C., Grossi, R., Pisanti, N. (eds) Combinatorial Algorithms. IWOCA 2019. Lecture Notes in Computer Science(), vol 11638. Springer, Cham. https://doi.org/10.1007/978-3-030-25005-8_15
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