Certified domination critical graphs upon vertex removal
Mateusz Miotk · Utilitas Mathematica · 2025
A set \(D\) of vertices of a graph \(G=(V_G,E_G)\) is a dominating set of \(G\) if every vertex in \(V_G-D\) is adjacent to at least one vertex in \(D\). The domination number of a graph \(G\), denoted by \(\gamma(G)\), is the cardinality of a smallest dominating set of \(G\). A subset \(D\subseteq V_G\) is called a certified dominating set of \(G\) if \(D\) is a dominating set of \(G\), and every vertex in \(D\) has either zero or at least two neighbours in \(V_G-D\). The cardinality of a smallest certified dominating set of \(G\) is called the certified domination number of \(G\), and it is denoted by \(\gamma_{\rm cer}(G)\). A vertex \(v\) of \(G\) is certified critical if \(\gamma_{\rm cer}(G -v) < \gamma_{\rm cer}(G)\), and a graph \(G\) is vertex certified domination critical or \(\gamma_{cer}\)-critical if the removal of any vertex reduces its certified domination number. In this paper, we give examples and properties of certified critical vertices and vertex certified domination critical graphs. As an example of an application of certified critical vertices, we give a constructive characterisation of trees for which the smaller partite set is a minimum certified dominating set.