Boundary graph classes for some maximum induced subgraph problems

The notion of a boundary graph class was recently introduced for a classification of hereditary graph classes according to the complexity of a considered problem. Two concrete graph classes are known to be boundary for several graph problems. We formulate a criterion to determine whether these classes are boundary for a given graph problem or not. We also demonstrate that the classes are simultaneously boundary for some continuous set of graph problems and they are not simultaneously boundary for another set of the same cardinality. Both families of problems are constituted by variants of the maximum induced subgraph problem.

[1]  Stephen T. Hedetniemi,et al.  A linear algorithm for disjoint matchings in trees , 1978, Discret. Math..

[2]  Frank Harary,et al.  Graph Theory , 2016 .

[3]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[4]  Brendan D. McKay,et al.  R(4, 5) = 25 , 1995, J. Graph Theory.

[5]  David S. Johnson,et al.  Some simplified NP-complete problems , 1974, STOC '74.

[6]  Celina M. H. de Figueiredo,et al.  SPLITTING NUMBER is NP-complete , 1998, Discret. Appl. Math..

[7]  W. Sierpinski,et al.  Sur le probléme des courbes gauches en Topologie , 2022 .

[8]  Vadim V. Lozin,et al.  Boundary properties of graphs for algorithmic graph problems , 2011, Theor. Comput. Sci..

[9]  Celina M. H. de Figueiredo,et al.  On maximum planar induced subgraphs , 2006, Discret. Appl. Math..

[10]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[11]  Vadim V. Lozin,et al.  Boundary classes of graphs for the dominating set problem , 2004, Discrete Mathematics.

[12]  V. E. Alekseev,et al.  On easy and hard hereditary classes of graphs with respect to the independent set problem , 2003, Discret. Appl. Math..

[13]  Reinhard Diestel,et al.  Graph Theory , 1997 .

[14]  Raffaele Mosca.  Some observations on maximum weight stable sets in certain P , 2008, Eur. J. Oper. Res..

[15]  Vadim V. Lozin,et al.  NP-hard graph problems and boundary classes of graphs , 2007, Theor. Comput. Sci..

[16]  D. Malyshev.  On the infinity of the set of boundary classes for the edge 3-colorability problem , 2010 .

[17]  P. Seymour,et al.  The Strong Perfect Graph Theorem , 2002, math/0212070.