# A linear-time algorithm for four-partitioning four-connected planar graphs

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## Abstract

Given a graph *G=(V, E)*, four distinct vertices *u*_{1},u_{2},u_{3},u_{4} ∈ V and four natural numbers *n*_{1}, n_{2}, n_{3}, n_{4} such that \(\sum\nolimits_{i = 1}^4 {n_i = |V|}\), we wish to find a partition *V*_{1}, V_{2}, V_{3}, V_{4} of the vertex set *V* such that *u*_{i} ∈ V_{i}, ¦V_{i}¦=n_{i} and *V*_{i} induces a connected subgraph of *G* for each *i, 1 ≤ i ≤ 4*. In this paper we give a simple linear-time algorithm to find such a partition if *G* is a 4-connected planar graph and *u*_{1}, u_{2}, u_{3}, u_{4} are located on the same face of a plane embedding of *G*. Our algorithm is based on a “4-canonical decomposition” of *G*, which is a generalization of an *st*-numbering and a “canonical 4-ordering” known in the area of graph drawings.

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## References

- [DF85]M.E. Dyer and A.M. Frieze,
*On the complexity of partitioning graphs into connected subgraphs*, Discrete Applied Mathematics, 10 (1985) 139–153.Google Scholar - [E79]S. Even,
*Graph Algorithms*, Computer Science Press, Potomac (1979).Google Scholar - [G78]E. Györi,
*On division of connected subgraphs*, Proc. 5th Hungarian Combinational Coll., (1978) 485–494.Google Scholar - [G96]E. Györi,
*Private communication*, March 21, 1996.Google Scholar - [JSN94]L. Jou, H. Suzuki and T. Nishizeki,
*A linear algorithm for finding a nonseparating ear decomposition of triconnected planar graphs*, Tech. Rep. of Information Processing Society of Japan, AL40-3 (1994).Google Scholar - [K94]G. Kant,
*A more compact visibility representation*, Proc. of the 19th International Workshop on Graph Theoretic Concepts in Computer Science (WG'93), LNCS 790 (1994) 411–424.Google Scholar - [KH94]G. Kant and X. He,
*Two algorithms for finding rectangular duals of planar graphs*, Proc. of the 19th International Workshop on Graph Theoretic Concepts in Computer Science (WG'93), LNCS 790 (1994) 396–410.Google Scholar - [L77]L. Lovász,
*A homology theory for spanning trees of a graph*, Acta Math. Acad. Sci. Hunger, 30 (1977) 241–251.Google Scholar - [MM94]J. Ma and S. H. Ma,
*An O(k*^{2}n^{2}) algorithm to find a k-partition in a k-connected graph, J. of Computer Sci. & Technol., 9, 1 (1994) 86–91.Google Scholar - [STN90]H. Suzuki, N. Takahashi and T. Nishizeki,
*A linear algorithm for bipartition of biconnected graphs*, Information Processing Letters 33, 5 (1990) 227–232.CrossRefGoogle Scholar - [STNMU90]H. Suzuki, N. Takahashi, T. Nishizeki, H. Miyano and S. Ueno,
*An algorithm for tripartitioning 3-connected graphs*, Journal of Information Processing Society of Japan 31, 5 (1990) 584–592.Google Scholar - [WK94]K. Wada and K. Kawaguchi,
*Efficient algorithms for triconnected graphs and 3-edge-connected graphs*, Proc. of the 19th International Workshop on Graph Theoretic Concepts in Computer Science (WG'93), LNCS 790 (1994) 132–143.Google Scholar - [WTK95]K. Wada, A. Takaki and K. Kawaguchi,
*Efficient algorithms for a mixed k-partition problem of graphs without specifying bases*, Proc. of the 20th International Workshop on Graph Theoretic Concepts in Computer Science (WG'94), LNCS 903 (1995) 319–330.Google Scholar

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