Given a complete graph $G$, an arbitrary graph $H$, and a positive integer $n$, are there subgraphs $A_1,\dots,A_n$ of $G$ (not necessarily disjoint) such that their union is $G$, and each of them are isomorphic to $H$?
This is a problem which I believe is NP-complete, but I am unsure if it is actually so. Any ideas of how to prove this?
- If $G$ is instead allowed to be an arbitrary graph, then this is clearly at least as hard as subgraph isomorphism, which is NP-complete.
- If $G$ is allowed to be an arbitrary graph, and $A_1,\dots,A_n$ are required to be isomorphic to a subgraph of $H$, instead of the entire graph $H$, then the problem can be reduced from vertex cover (which is NP-complete) by setting $H$ to be a star graph with the number of spokes equal to the number of nodes in $G$.
- The problem where $H$ must also be a complete graph is called "covering design", and has some discussion here, with closed forms for when $H$ has 3 or 4 nodes. Finding algorithm for this is apparently an open problem.