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Packing [1, Δ]-factors in graphs of small degree

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Abstract

Given an undirected, connected graph G with maximum degree Δ, we introduce the concept of a [1, Δ]-factor k-packing in G, defined as a set of k edge-disjoint subgraphs of G such that every vertex of G has an incident edge in at least one subgraph. The problem of deciding whether a graph admits a [1,Δ]-factor k-packing is shown to be solvable in linear time for k = 2, but NP-complete for all k≥ 3. For k = 2, the optimisation problem of minimising the total number of edges of the subgraphs of the packing is NP-hard even when restricted to subcubic planar graphs, but can in general be approximated within a factor of \(\frac{42\Delta -30}{35\Delta-21}\) by reduction to the Maximum 2-Edge-Colorable Subgraph problem. Finally, we discuss implications of the obtained results for the problem of fault-tolerant guarding of a grid, which provides the main motivation for research.

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References

  • Caro Y (1984) The decomposition of trees into subtrees. J Graph Theor 8:471–479

    MATH  MathSciNet  Google Scholar 

  • de Castro N, Cobos FJ, Dana JC, Marquez A, Noy M (1999) Triangle-free planar graphs as segment intersection graphs. Lecture Notes Comp Sci 1731:341–358

    Google Scholar 

  • Chazelle B, Edelsbrunner H (1992) An optimal algorithm for intersecting line segments in the plane. J ACM 39:1–54

    Article  MATH  MathSciNet  Google Scholar 

  • Colbourn CJ (1988) Edge-packing of graphs and network ealiability. Disc Math 72:49–61

    Article  MATH  MathSciNet  Google Scholar 

  • Czyzowicz J, Kranakis E, Urrutia J (1998) A simple proof of the representation of bipartite planar graphs as the contact graphs of orthogonal straight line segments. Inform Proc Lett 66:125–126

    Article  MATH  MathSciNet  Google Scholar 

  • Dangelmayr C (2004) Intersection graphs of straight-line segments. In: 4th Annual CGC Workshop on Computational Geometry (Stels)

  • Dyer ME, Frieze AM (1986) Planar 3DM is NP-complete. J Algorith 7(1):174–184

    Article  MATH  MathSciNet  Google Scholar 

  • Favrholdt LM, Nielsen MN (2003) On-line edge-coloring with a fixed number of colors. Algorithmica 35(2):176–191

    Google Scholar 

  • Finke U, Hinchirs K (1995) Overlaying simply connected planar subdivisions in linear time. In: Proc ACM SoCG, pp 119–126

  • Feige U, Ofek E, Wieder U (2002) Approximating maximum edge coloring in multigraphs. Lecture Notes Comp Sci 2462:108–121

    Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and intractability: A guide to the theory of NP-completeness. Freeman, New York

    MATH  Google Scholar 

  • Hartman IB-A, Newman I, Ziv R (1991) On grid intersection graphs. Disc Math 87:41–52

    Article  MATH  MathSciNet  Google Scholar 

  • Hartvigsen D (1984) Extensions of matching theory, PhD Thesis. Carnegie-Mellon University

  • Heath LS, Vergara JPC (1998) Edge-packing planar graphs by cyclic graphs. Disc Appl Math 81(1–3):169–180

    Article  MATH  MathSciNet  Google Scholar 

  • Holyer I (1981) The NP-completeness of edge-colouring. SIAM J Comp 10:718–720

    Article  MATH  MathSciNet  Google Scholar 

  • Ivanco J, Meszka M, Skupień Z (2002) Decompositions of multigraphs into parts with two edges. Discuss Math Graph Theor 22:113–121

    MATH  Google Scholar 

  • Jacobs DP, Jamison RE (2000) Complexity of recognizing equal unions in families of sets. J Algor 37(2):495–504

    Article  MATH  MathSciNet  Google Scholar 

  • Jünger M, Reinelt G, Pulleyblank WR (1985) On partitioning the edges of graphs into connected subgraphs. J Graph Theor 9:539–549

    MATH  Google Scholar 

  • Kawarabayashi K, Matsuda H, Oda Y, Ota K (2002) Path factors in cubic graphs. J Graph Theor 39:188–193

    Article  MathSciNet  Google Scholar 

  • König D (1916) Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Math Ann 77:453–465

    Article  MathSciNet  Google Scholar 

  • Kosowski A, Małafiejski M, Żyliński P (in press) Cooperative mobile guards in grids, to appear in Computational Geometry: Theory and Applications

  • Kosowski A, Małafiejski M, Żyliński P (2006) Fault tolerant guarding of grids. Lecture Notes Comp Sci 3980:161–170

    Google Scholar 

  • Leven D, Galil Z (1983) NP-completeness of finding the chromatic index of regular graphs. J Algorith 4(1):35–44

    Article  MATH  MathSciNet  Google Scholar 

  • Liaw BC, Huang NF, Lee RCT (1993) The minimum cooperative guards problem on k-spiral polygons. In: Proc CCCG, pp 97–101

  • Liaw BC, Lee RCT (1994) An optimal algorithm to solve the minimum weakly cooperative guards problem for 1-spiral polygons. Inf Process Lett 52:69–75

    Article  MATH  MathSciNet  Google Scholar 

  • Lonc Z, Meszka M, Skupień Z (2004) Edge decompositions of multigraphs into 3-matchings. Graphs Comb 20:507–515

    Article  MATH  Google Scholar 

  • Małafiejski M, Żyliński P (2005) Weakly cooperative guards in grids. Lecture Notes Comp Sci 3480:647–656

    Google Scholar 

  • Nierhoff T, Żyliński P (2003) Cooperative guards in grids. In: 3rd Annual CGC Workshop on Computational Geometry, Neustrelitz

  • Ntafos S (1986) On gallery watchmen in grids. Inf Proc Lett 23:99–102

    Article  MATH  Google Scholar 

  • O’Rourke J (1987) Art gallery theorems and algorithms. Oxford University Press

  • Scheinerman ER (1984) Intersection classes and multiple intersection parameters of graphs. PhD thesis, Princenton University

  • Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  • Urrutia J (2000) Art gallery and illumination problems. Handbook on computational geometry, Elsevier Science, Amsterdam

    Google Scholar 

  • Woźniak M (2004) Packing of graphs and permutations—a survey. Disc Math 276(1–3):379–391

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Algorithms and System Modeling, Gdańsk University of Technology, 80-952, Gdańsk, Poland

    Adrian Kosowski & Michał Małafiejski

  2. Institute of Mathematics, University of Gdańsk, 80-952, Gdańsk, Poland

    Paweł Żyliński

Authors
  1. Adrian Kosowski
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  2. Michał Małafiejski
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  3. Paweł Żyliński
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Correspondence to Paweł Żyliński.

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Kosowski, A., Małafiejski, M. & Żyliński, P. Packing [1, Δ]-factors in graphs of small degree. J Comb Optim 14, 63–86 (2007). https://doi.org/10.1007/s10878-006-9034-4

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  • DOI: https://doi.org/10.1007/s10878-006-9034-4

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