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

Adrian Kosowski, Michał Małafiejski and Paweł Żyliński ()
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Adrian Kosowski: Gdańsk University of Technology
Michał Małafiejski: Gdańsk University of Technology
Paweł Żyliński: University of Gdańsk

Journal of Combinatorial Optimization, 2007, vol. 14, issue 1, No 5, 63-86

Abstract: 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.

Keywords: Packing factors; Edge-colorable subgraph; Fault tolerant guards (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10878-006-9034-4

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