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An extended tree of a graph is a certain analogue of spanning tree. It is defined by means of vertex splitting. The properties of these trees are studied, mainly for complete graphs.
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Gary Chartrand, Ortrud R. Oellermann, Song Lin Tian, Hung Bin Zou (1989)
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For a fixed graph F, a graph G is F-saturated if there is no copy of F in G, but for any edge e ∉ G, there is a copy of F in G + e. The minimum number of edges in an F-saturated graph of order n will be denoted by sat(n, F). A graph G is weakly F-saturated if there is an ordering of the missing edges of G so that if they are added one at a time, each edge added creates a new copy of F. The minimum size of a weakly F-saturated graph G of order n will be denoted by wsat(n, F). The graphs...