Abstract
The vertex-disjoint triangles (VDT) problem asks for a set of maximum number of pairwise vertex-disjoint triangles in a given graph G. The triangle cover problem asks for the existence of a perfect triangle packing in a graph G. It is known that the triangle cover problem is NP-complete on general graphs with clique number 3 [6]. The VDT problem is MAX SNP-hard on graphs with maximum degree four, while it can be approximated within 3/2+ε, for any ε > 0, in polynomial time [11].
We prove that the VDT problem is NP-complete even when the input graphs are chordal, planar, line or total graphs. We present an \(O(m \sqrt{n})\) algorithm for the VDT problem on split graphs and an O(n 3) algorithm for the VDT problem on cographs. A linear algorithm for the triangle cover problem on strongly chordal graphs is also presented. Finally, the notion of packing-hardness, which may be crucial to the understanding of the difficulty of generalized matching problems, is defined.
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Guruswami, V., Rangan, C.P., Chang, M.S., Chang, G.J., Wong, C.K. (1998). The Vertex-Disjoint Triangles Problem. In: Hromkovič, J., Sýkora, O. (eds) Graph-Theoretic Concepts in Computer Science. WG 1998. Lecture Notes in Computer Science, vol 1517. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10692760_3
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DOI: https://doi.org/10.1007/10692760_3
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