For a property π on graphs, the edge-contraction problem with respect to π is defined as a problem of finding a set of edges of minimum cardinality whose contraction results in a graph satisfying the property π. This paper gives a lower bound for the approximation ratio for the problem for any property π that is hereditary on contractions and determined by biconnected components.

The minimum biclique edge cover problem (MBECP) is NP-hard for general graphs. It is known that if we restrict an input graph to the bipartite domino-free class, MBECP can be solved within polynomial-time of input graph size. We show a new polynomial-time solvable graph class for MBECP that is characterized by three forbidden graphs, a domino, a gem and K4. This graph class allows that an input graph is non-bipartite, and includes the bipartite domino-free graph class properly.

For a graph G, a biclique edge partition SBP(G) is a collection of bicliques (complete bipartite subgraphs) Bi such that each edge of G is contained in exactly one Bi. The Minimum Biclique Edge Partition Problem (MBEPP) asks for SBP(G) with the minimum size. In this paper, we show that for arbitrary small ε>0, (6053/6052-ε)-approximation of MBEPP is NP-hard.

The minimum biclique cover problem is known to be NP-hard for general bipartite graphs. It can be solved in polynomial time for C4-free bipartite graphs, bipartite distance hereditary graphs and bipartite domino-free graphs. In this paper, we define the modified Galois lattice Gm(B) for a bipartite graph B and introduce the redundant parameter R(B). We show that R(B)=0 if and only if B is domino-free. Furthermore, for an input graph such that R(B)=1, we show that the minimum biclique cover problem can be solved in polynomial time.