General Bounds on the Downhill Domination Number in Graphs.
A path π = (v1, v2,...vk+1) in a graph G = (V,E) is a downhill path if for every i, 1 < i < k, deg(vi) > deg(vi+1), where deg(vi) denotes the degree of vertex vi ∊ V. The downhill domination number equals the minimum cardinality of a set S ⊂ V having the property that every vertex v ∊ V lie...
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| Format: | Others |
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Digital Commons @ East Tennessee State University
2013
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| Online Access: | https://dc.etsu.edu/honors/107 https://dc.etsu.edu/cgi/viewcontent.cgi?article=1111&context=honors |
| Summary: | A path π = (v1, v2,...vk+1) in a graph G = (V,E) is a downhill path if for every i, 1 < i < k, deg(vi) > deg(vi+1), where deg(vi) denotes the degree of vertex vi ∊ V. The downhill domination number equals the minimum cardinality of a set S ⊂ V having the property that every vertex v ∊ V lies on a downhill path originating from some vertex in S. We investigate downhill domination numbers of graphs and give upper bounds. In particular, we show that the downhill domination number of a graph is at most half its order, and that the downhill domination number of a tree is at most one third its order. We characterize the graphs obtaining each of these bounds. |
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