The k-piece packing problem

A k-piece of a graph G is a connected subgraph of G all of whose nodes have degree at most k and at least one node has degree equal to k. We consider the problem of covering the maximum number of nodes of a graph by node disjoint k-pieces. When k = 1 this is the maximum matching problem, and when k = 2 this is the problem, recently studied by Kaneko [19[, of covering the maximum number of nodes by disjoint paths of length greater than 1. We present a polynomial time algorithm for the problem as well as a Tutte-type existence theorem and a Berge-type min-max formula. We also solve the problem in the more general situation where the “pieces” are defined in terms of lower and upper bounds on the degrees. © 2006 Wiley Periodicals, Inc. J Graph Theory

[1]  László Lovász,et al.  Subgraphs with prescribed valencies , 1970 .

[2]  Richard C. Brewster,et al.  Packing paths in digraphs , 2003 .

[3]  Jácint Szabó,et al.  The Edmonds-Gallai Decomposition for the k-Piece Packing Problem , 2005, Electron. J. Comb..

[4]  Gérard Cornuéjols,et al.  Critical graphs, matchings and tours or a hierarchy of relaxations for the travelling salesman problem , 1983 .

[5]  Martin Loebl,et al.  Efficient Subgraph Packing , 1993, J. Comb. Theory, Ser. B.

[6]  Hong Wang,et al.  Path factors of bipartite graphs , 1994, J. Graph Theory.

[7]  Michel Las Vergnas.  An extension of Tutte's 1-factor theorem , 1978, Discret. Math..

[8]  David G. Kirkpatrick,et al.  Packings by cliques and by finite families of graphs , 1984, Discret. Math..

[9]  Richard C. Brewster,et al.  On the complexity of digraph packings , 2003, Inf. Process. Lett..

[10]  Atsushi Kaneko,et al.  A necessary and sufficient condition for the existence of a path factor every component of which is a path of length at least two , 2003, J. Comb. Theory, Ser. B.

[11]  Michel Mollard.  A generalized parity function and its use in the construction of perfect codes , 1986 .

[12]  Harold N. Gabow,et al.  An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems , 1983, STOC.

[13]  Gérard Cornuéjols,et al.  An extension of matching theory , 1986, J. Comb. Theory, Ser. B.

[14]  David G. Kirkpatrick,et al.  On the Complexity of General Graph Factor Problems , 1981, SIAM J. Comput..

[15]  Zoltán Király,et al.  Packing paths of length at least two , 2004, Discret. Math..

[16]  W. T. Tutte.  The Factorization of Linear Graphs , 1947 .

[17]  Martin Loebl,et al.  On matroids induced by packing subgraphs , 1988, J. Comb. Theory, Ser. B.