The complexity of the Pk partition problem and related problems in bipartite graphs

In this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems

The complexity of the Pk partition problem and related problems in bipartite graphs

International audienceIn this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems

A necessary and sufficient condition for the existence of a path factor every component of which is a path of length at least two

AbstractA P⩾3-factor F of a graph G is a spanning subgraph of G such that every component of F is a path of length at least two. Let R be a factor-critical graph with at least three vertices, that is, for each x∈V(R),R−x has a 1-factor (i.e., a perfect matching). Set V(R)={x1,…,xn}. Add new vertices {v1,…,vn} to R together with the edges xivi,1⩽i⩽n. The resulting graph H is called a sun. (Note that degHvi=1 for all i,1⩽i⩽n.) K1 and K2, i.e., the complete graphs with one and two vertices, respectively, are also called suns. Then let C be the set of all suns. A sun component of a graph is a component which belongs to C. Let cs(G) denote the number of sun components of G. We prove that a graph G has a P⩾3-factor if and only if cs(G−S)⩽2|S|, for every subset S of V(G)