On Toughness and Hamiltonicity of 2K2-Free Graphs


  • Contract grant sponsor: EPSRC; Contract grant number: EP/F064551/1; Contract grant sponsor: RFBR; Contract numbers: 12-01-00090, 12-01-00093, 12-01-00184, and 12-01-33028; Contract grant sponsor: Ministry of Education and Science of the Russian Federation; Contract grant number: 14.740.11.0868.


The toughness of a (noncomplete) graph G is the minimum value of t for which there is a vertex cut A whose removal yields math formula components. Determining toughness is an NP-hard problem for general input graphs. The toughness conjecture of Chvátal, which states that there exists a constant t such that every graph on at least three vertices with toughness at least t is hamiltonian, is still open for general graphs. We extend some known toughness results for split graphs to the more general class of 2K2-free graphs, that is, graphs that do not contain two vertex-disjoint edges as an induced subgraph. We prove that the problem of determining toughness is polynomially solvable and that Chvátal's toughness conjecture is true for 2K2-free graphs.