A complete and equal computational complexity classification of compaction and retraction to all graphs with at most four vertices and some general results

A very close relationship between the compaction, retraction, and constraint satisfaction problems has been established earlier providing evidence that it is likely to be difficult to give a complete computational complexity classification of the compaction and retraction problems for reflexive or bipartite graphs. In this paper, we give a complete computational complexity classification of the compaction and retraction problems for all graphs (including partially reflexive graphs) with four or fewer vertices. The complexity classification of both the compaction and retraction problems is found to be the same for each of these graphs. This relates to a long-standing open problem concerning the equivalence of the compaction and retraction problems. The study of the compaction and retraction problems for graphs with at most four vertices has a special interest as it covers a popular open problem in relation to the general open problem. We also give complexity results for some general graphs. The compaction and retraction problems are special graph colouring problems, and can also be viewed as partition problems with certain properties. We describe some practical applications also.

[1]  NARAYAN VIKAS,et al.  Computational Complexity of Compaction to Reflexive Cycles , 2002, SIAM J. Comput..

[2]  Pavol Hell,et al.  Absolute retracts in graphs , 1974 .

[3]  Richard J. Nowakowski,et al.  Fixed-edge theorem for graphs with loops , 1979, J. Graph Theory.

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

[5]  P. Hell,et al.  GRAPHS WITH FORBIDDEN HOMOMORPHIC IMAGES , 1979 .

[6]  Narayan Vikas,et al.  Computational complexity of compaction to cycles , 1999, SODA '99.

[7]  Sulamita Klein,et al.  List Partitions , 2003, SIAM J. Discret. Math..

[8]  Pavol Hell,et al.  List Homomorphisms to Reflexive Graphs , 1998, J. Comb. Theory, Ser. B.

[9]  Sulamita Klein,et al.  Complexity of graph partition problems , 1999, STOC '99.

[10]  Hans-Jürgen Bandelt,et al.  Absolute retracts of bipartite graphs , 1987, Discret. Appl. Math..

[11]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[12]  Erwin Pesch,et al.  A characterization of absolute retracts of n-chromatic graphs , 1985, Discret. Math..

[13]  Y VardiMoshe,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction , 1999 .

[14]  Pavol Hell,et al.  List Homomorphisms and Circular Arc Graphs , 1999, Comb..

[15]  Frank Thomson Leighton,et al.  A Graph Coloring Algorithm for Large Scheduling Problems. , 1979, Journal of research of the National Bureau of Standards.

[16]  P. Hell,et al.  Absolute Retracts and Varieties of Reflexive Graphs , 1987, Canadian Journal of Mathematics.

[17]  Michael Randolph Garey,et al.  An Application of Graph Coloring to Printed Circuit Testing (Working Paper) , 1975, FOCS 1975.

[18]  Narayan Vikas,et al.  Computational complexity of compaction to irreflexive cycles , 2004, J. Comput. Syst. Sci..

[19]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[20]  Hans-Jürgen Bandelt,et al.  Absolute Reflexive Retracts and Absolute Bipartite Retracts , 1993, Discret. Appl. Math..

[21]  David S. Johnson,et al.  An application of graph coloring to printed circuit testing , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[22]  Jaroslav Nesetril,et al.  On the complexity of H-coloring , 1990, J. Comb. Theory, Ser. B.

[23]  Tomás Feder,et al.  Monotone monadic SNP and constraint satisfaction , 1993, STOC.

[24]  Narayan Vikas,et al.  Computational Complexity Classification of Partition under Compaction and Retraction , 2004, COCOON.

[25]  Narayan Vikas,et al.  Compaction, Retraction, and Constraint Satisfaction , 2004, SIAM J. Comput..

[26]  Derick Wood,et al.  Colorings and interpretations: a connection between graphs and grammar forms , 1981, Discret. Appl. Math..