Domination and independence subdivision numbers of graphs (Q2725191)

A subset \(S\) of the vertex set \(V(G)\) of a graph \(G\) is called dominating in \(G\), if each vertex of \(G\) either is in \(S\), or is adjacent to a vertex of \(S\). A set \(S\subseteq V(G)\) is independent in \(G\), if no two vertices of \(S\) are adjacent in \(G\). The minimum number of vertices of a dominating set in \(G\) is the domination number \(\gamma(G)\) of \(G\), the maximum number of vertices of an independent set in \(G\) is the independence number \(\beta(G)\) of \(G\). Arumugam has introduced the concept of domination subdivision number \(\text{sd}_\gamma(G)\). This is the minimum number of edges which must be subdivided (everyone only once) in order to increases \(\gamma(G)\). Analogously the authors introduce the independence subdivision number \(\text{sd}_\beta(G)\) as the minimum number of edges which must be subdivided (again everyone only once) in order to increase \(\beta(G)\). In the paper it is proved that, for any connected graph \(G\) with \(n\geq 3\) vertices and for any two adjacent vertices \(u\), \(v\) in it with degrees at least 2, the inequality \(\text{sd}_\gamma(G)\leq \text{deg}(u)+ \text{deg}(v)- 1\) holds. This theorem implies two corollaries, for grid graphs and for regular graphs (in general). Further it is proved that \(\text{sd}_\beta(G)= m\) for \(G\) being the star with \(m\) edges and otherwise \(1\leq \text{sd}_\beta(G)\leq 2\). The graphs \(G\) with \(\text{sd}_\beta(G)= 2\) are characterized.