Isolation subdivision number of a graph

Abstract

For a graph $G=(V,E),$ a set $S\subseteq V$ is called an isolating set of $G$ if the set $V-N[S]$ is independent. The minimum cardinality of an isolating set in $G$ is the isolation number of $G$, denoted by ${\hat{I}}^1(G).$ Here we introduce the isolation subdivision number of a graph $G$, denoted by ${\mathrm{sd}}_{\hat{I}}^1(G)$, as the minimum number of edges of $G$ that must be subdivided, where each edge can be subdivided at most once, in order to obtain a graph with isolation number greater than ${\hat{I}}^1(G).$ We show that the new parameter is well defined for any non-trivial graph different from a star and that it can be arbitrarily large. We present the values of this parameter for some elementary classes of graphs and establish some basic properties. We show also that $1\le {\mathrm{sd}}_{\hat{I}}^1(T)\le 4$ for any tree $T$ different from a star and characterize all trees $T$ with ${\mathrm{sd}}_{\hat{I}}^1(T)=1.$

15 pages, 3 figures, 35 references