Abstract
Graph Theory Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove that the biclique vertex-partition problem is polynomially solvable for bipartite permutation graphs, bipartite distance-hereditary graphs and remains NP-complete for perfect elimination bipartite graphs and bipartite graphs containing no 4-cycles as induced subgraphs.
Highlights
In this paper we consider the computational complexity of the problem of partitioning the vertex set of the given graph into at most k bicliques (BICLIQUE VERTEX-PARTITION) in subclasses of bipartite graphs
The aim of this paper is to investigate the following question: which subclasses of bipartite graphs does the BICLIQUE VERTEX-PARTITION problem become polynomially solvable for? In Section 3 we establish a relation between the considered problem restricted to (K3, C5, C6)-free graphs and colouration problems of graphs
We study the computational complexity of BICLIQUE VERTEX-PARTITION for perfect elimination bipartite graphs
Summary
In this paper we consider the computational complexity of the problem of partitioning the vertex set of the given graph into at most k bicliques (BICLIQUE VERTEX-PARTITION) in subclasses of bipartite graphs. It is known that the BICLIQUE VERTEX-PARTITION problem is NP-complete for bipartite graphs [8, 11] and polynomially solvable for trees [1]. The aim of this paper is to investigate the following question: which subclasses of bipartite graphs does the BICLIQUE VERTEX-PARTITION problem become polynomially solvable for? We establish the NP-completeness of this problem for perfect elimination bipartite graphs and bipartite C4-free graphs
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