On the planar, outer planar, cut vertices and end-regular comaximal graph of lattices (Q2834156)

Given a ring \(R\), the comaximal graph of \(R\), denoted by \(\Gamma (R)\), has the vertex set \(V(\Gamma) := R\) and \(\{a, b\} \in E(\Gamma)\) iff \(Ra + Rb = R\). If \((L, \wedge, \vee)\) is a lattice, the comaximal graph of \(L\), denoted by \(\Gamma (L)\), has the vertex set \(V(\Gamma):= L\) and \(\{a, b\} \in E(\Gamma)\) iff \(a \vee b = 1\). If, in \(\Gamma(L)\), one removes the vertex corresponding to \(1 \in L\) and all isolated vertices, the induced subgraph of \(\Gamma(L)\) is denoted by \(\Gamma_2(L)\).NEWLINENEWLINE The paper provides conditions for:NEWLINE{\parindent=0.7cm\begin{itemize}\item[(1)] a vertex to be a cut vertex in \(\Gamma_2(L)\);NEWLINE\item[(2)] planarity of \(\Gamma_2(L_1 \times L_2)\);NEWLINE\item[(3)] outerplanarity of \(\Gamma_2(L)\) and of \(\Gamma_2(L_1 \times L_2)\);NEWLINE\item[(4)] end-regularity of \(\Gamma_2(L)\) and of \(\Gamma_2(L_1 \times L_2)\).\end{itemize}}