Section: New Results

On the minimum edge cover and vertex partition by quasi-cliques problems

A $\gamma$-quasi-clique in a simple undirected graph is a set of vertices which induces a subgraph with the edge density of at least $\gamma$ for $0<\gamma <1$. A cover of a graph by $\gamma$-quasi-cliques is a set of $\gamma$-quasi-cliques where each edge of the graph is contained in at least one quasi-clique. The minimum cover by $\gamma$-quasi-cliques problem asks for a $\gamma$-quasi-clique cover with the minimum number of quasi-cliques. A partition of a graph by $\gamma$-quasi-cliques is a set of $\gamma$-quasi-cliques where each vertex of the graph belongs to exactly one quasi-clique. The minimum partition by $\gamma$-quasi-cliques problem asks for a vertex partition by $\gamma$-quasi-cliques with the minimum number of quasi-cliques. In this work [60] , we show that the decision versions of the minimum cover and partition by $\gamma$-quasi-cliques problems are NP-complete for any fixed $\gamma$ satisfying $0<\gamma <1$.