| Summary: | 碩士 === 銘傳大學 === 資訊傳播工程學系碩士班 === 104 === A minus (respectively, signed) dominating function of a graph G = (V, E) is a function f：V → {-1, 0, 1} (respectively, {-1, 1}) such that Σu∈N_G[v] f(u) ≥ 1 for all v ∈ V , where N_G [v] = {v}⋃{u|(u, v) ∈ E}. The weight of a minus (respectively, signed) dominating function of G is the sum of its function values over all vertices. The minus respectively, signed) domination problem is to find a minus (respectively, signed) dominating function of G of minimum weight. In this thesis, we study some variations of the signed domination and minus domination problems such as the signed k-domination, reverse signed domination, and reverse minus domination problems. The results of the thesis are as follows：
Let k be a fixed nonnegative integer. For doubly chordal graphs and bipartite planar graphs, we show that the signed k-domination problem is NP-complete. We also show that the signed k-domination problem is not fixed parameter tractable.
For chordal graphs and bipartite planar graphs, we show that the reverse minus domination problem is NP-complete. For doubly chordal graphs and bipartite planar graphs, we show that the reverse signed domination problem is NP-complete. Furthermore, we show that even when restricted to bipartite planar graphs or doubly chordal graphs, the reverse signed domination problem is not fixed parameter tractable.
For strongly chordal graphs and distance-hereditary graphs, we show that the signed k-domination, reverse signed domination, and reverse minus domination problems can be solved in polynomial time. We also show that these three problems are linear-time solvable for trees, interval graphs, and chordal comparability graphs.
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