Upper Domination Parameters and Edge-Critical Graphs

Abstract

For $\pi$ one of the upper domination parameters $\beta$, $\mathrm{\Gamma }$, or $IR$, we investigate graphs for which $\pi$ decreases ( $\pi$-edge-critical graphs) and graphs for which $\pi$ increases ( ${\pi }^{+}$-edge-critical graphs) whenever an edge is added. We find characterisations of $\beta$- and $\mathrm{\Gamma }$-edge-critical graphs and show that a graph is $IR$-edge-critical if and only if it is $\mathrm{\Gamma }$-edge-critical. We also exhibit a class of ${\mathrm{\Gamma }}^{+}$-edge-critical graphs.