Independent domination critical graphs
Date
1994
Authors
Ao, Suqin
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Abstract
The domination number of a graph 𝘎 is the minimum size of any dominating set of 𝘎. The independent domination number of 𝘎 is the minimum size of any independent dominating set of 𝘎.
We study several different definitions of graphs which are critical with respect to the domination number (respectively, independent domination number), and give examples of infinite families of graphs of each type. Comparing these families leads to four principal types of critical graphs, that is, graphs for which the delet10n of any vertex causes the domination number (respectively, the independent domination number) to decrease, and graphs for which adding an edge causes the domination number (respectively, the independent domination number) to decrease. Since graphs for which such changes cause the domination number to decrease have been studied in the literature, we concentrate on those for which such changes cause the independent domination number to decrease. In each case we develop the theory of independent domination critical graphs which is analogous to that, already in the literature, for domination critical graphs. We extend this theory, and that for domination critical graphs, by giving a complete classification of the vertex domination critical graphs with the maximum possible diameter. We also explore Hamiltonian properties of edge independent domination critical graphs by proving that every 2-connected edge independent domination critical graph with independent domination number three has a Hamilton cycle. We then develop a complete class1ficat10n of edge independent domination critical graphs with a cut-vertex. This leads to the result that any connected edge independent domination critical graph with more than six vertices has a Hamilton path.