A factoring of a graph G = (V, E) is a collection of spanning subgraphs F1, F2, ... , Fk, 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 Fi is itself a graph and so has a dominating set. This set is called a local dominating set or LDS. An LDS of minimum size contains γi 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 γ nodes.
A central question answered by this dissertation is under what circumstances, given a set of integers γl, γ2, ... , γk, and γ, there is a graph which can be factored into k factors in such a way that a minimum LDS of Fi has size γl, 1 ≤ i ≤ k, and a GDS has size γ.
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 1s 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.
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Brigham, Robert C.
Doctor of Philosophy (Ph.D.)
College of Arts and Sciences
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Doctoral Dissertation (Open Access)
Carrington, Julie R., "Global Domination of Factors of a Graph" (1992). Retrospective Theses and Dissertations. 4389.
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