Let G be a group. The co-maximal graph of subgroups of G, denoted by \(\Gamma (G)\) , is a graph whose vertices are proper subgroups of G, and two distinct vertices H and K are adjacent if and only if \(HK=G\) . The deleted co-maximal subgroup graph of G, denoted by \(\Gamma ^{*}(G)\) , is defined as the graph obtained by removing the isolated vertices from \(\Gamma (G)\) . In this paper, first we study connectivity, diameter, girth, domination number and some other properties of the deleted co-maximal subgroup graphs of some groups. Then, from the point of view of the number of maximal subgroups in an abelian group G, we give necessary and sufficient conditions for the deleted co-maximal subgroup graph \(\Gamma ^{*}(G)\) to be planar. Finally, we establish a sufficient condition for the graph \(\Gamma ^{*}(G)\) to be perfect.
