Combinatorial Algorithms

Maximum Common Induced Subgraph (henceforth MCIS) is among the most studied classical NP-hard problems. MCIS remains NP-hard on many graph classes including bipartite graphs, planar graphs and k-trees. Little is known, however, about the parameterized complexity of the problem. When parameterized by the vertex cover number of the input graphs, the problem was recently shown to be fixed-parameter tractable. Capitalizing on this result, we show that the problem does not have a polynomial kernel when parameterized by vertex cover unless NP ⊆ coNP/poly. We also show that Maximum Common Connected Induced Subgraph (MCCIS), which is a variant where the solution must be connected, is also fixed-parameter tractable when parameterized by the vertex cover number of input graphs. Both problems are shown to be W[1]-complete on bipartite graphs and graphs of girth five and, unless P = NP, they do not belong to the class XP when parameterized by a bound on the size of the minimum feedback vertex sets of the input graphs, that is solving them in polynomial time is very unlikely when this parameter is a constant.

[1]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[2]  Sylwia Cichacz.  Distance magic graphs G × C n , 2014 .

[3]  Robin Thomas,et al.  Tutte's Edge-Colouring Conjecture , 1997, J. Comb. Theory, Ser. B.

[4]  Robin Thomas,et al.  Efficiently four-coloring planar graphs , 1996, STOC '96.

[5]  Udi Manber,et al.  Fast text searching: allowing errors , 1992, CACM.

[6]  Diana Combe,et al.  Magic labellings of graphs over finite abelian groups , 2004, Australas. J Comb..

[7]  Dalibor Froncek,et al.  Group distance magic labeling of Cartesian product of cycles , 2013, Australas. J Comb..

[8]  Robin Thomas,et al.  Recent Excluded Minor Theorems , 2022 .

[9]  J. Spencer.  Minimal scrambling sets of simple orders , 1972 .

[10]  Joseph A. Gallian,et al.  A Dynamic Survey of Graph Labeling , 2009, The Electronic Journal of Combinatorics.

[11]  Lukasz Kowalik.  Improved Edge-Coloring with Three Colors , 2006, WG.

[12]  Dalibor Froncek,et al.  Handicap distance antimagic graphs and incomplete tournaments , 2013 .

[13]  Bruce A. Reed,et al.  The disjoint paths problem in quadratic time , 2012, J. Comb. Theory, Ser. B.

[14]  Yoshio Okamoto,et al.  Counting the Number of Matchings in Chordal and Chordal Bipartite Graph Classes , 2009, WG.

[15]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[16]  Sylwia Cichacz-Przenioslo,et al.  Distance magic circulant graphs , 2016, Discret. Math..

[17]  Peter J. Slater.  A constructive characterization of trees with at least k disjoint maximum matchings , 1978, J. Comb. Theory, Ser. B.

[18]  Sylwia Cichacz.  Note on group distance magic complete bipartite graphs , 2013 .