Global domination of factors of a graph

TLDR: A strong relationship is demonstrated between two application areas and the ideas of global domination and factoring and it is found that a factoring of a graph can represent the parallel computation of a class of constraint problems or the routes of multicast messages in a network.

Abstract: A factoring of a graph G = (V, E) is a collection of spanning subgraphs F$\sb1$,F$\sb2,$ ..., F$\sb{\rm k}$, known as factors into which the edge set E has been partitioned. A dominating set of a graph is a set of nodes such that every node in the graph is either contained in the set or has an edge to some node in the set. Each factor F$\sb{\rm i}$ is itself a graph and so has a dominating set. This set is called a local dominating sets or LDS. An LDS of minimum size contains $\gamma\sb{\rm i}$ nodes. In addition, there is some set of nodes named a global dominating set which dominates all of the factors. If a global dominating set is of minimum size, it is called a GDS and contains $\gamma$ nodes. A central question answered by this dissertation is under what circumstances, given a set of integers $\gamma\sb1,\gamma\sb2,\...,\gamma\sb{\rm k}$, and $\gamma$, there is a graph which can be factored into k factors in such a way that a minimum LDS of F$\sb{\rm i}$ has size $\gamma\sb{\rm i},$ 1 $\le$ i $\le$ k, and a GDS has size $\gamma$. The general solution to this central question is complicated. In addition, simpler subproblems are often precisely those which are most applicable to practical problems. For these reasons, simpler solutions are found for several special cases of the general characterization problem. A strong relationship is demonstrated between two application areas and the ideas of global domination and factoring. We find that a factoring of a graph can represent the parallel computation of a class of constraint problems or the routes of multicast messages in a network. The applicability of these ideas is limited by the computational complexity of the problem of finding a GDS in a factoring. The problem is NP-Hard in general and we find that, more surprisingly, when the factors are very simple structures such as trees or even paths, the problem remains NP-Hard.