Isolation critical graphs under multiple edge subdivision

Abstract

This paper introduces the notion of an $({\hat{I}}^1,q)$-critical graph. The isolation number of a graph $G$, denoted by ${\hat{I}}^1(G)$ and also known as the vertex-edge domination number of $G$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the subgraph induced by the set of vertices that are not in the closed neighbourhood of $D$ has no edges. A graph $G$ is $({\hat{I}}^1,q)$-critical if every subdivision of $q$ edges of $G$ gives a graph whose isolation number is greater than ${\hat{I}}^1(G)$, and $G$ has $q-1$ edges such that subdividing them gives a graph whose isolation number is ${\hat{I}}^1(G)$. We show that an $({\hat{I}}^1,q)$-critical graph exists for every integer $q\ge 1$. We prove that if $G$ is a connected $m$-edge non-star graph, then $G$ is $({\hat{I}}^1,q)$-critical for some $q\le m-1$. We show that this bound is best possible. We provide a general characterization of $({\hat{I}}^1,1)$-critical graphs as well as a constructive characterization of $({\hat{I}}^1,1)$-critical trees, demonstrating that $({\hat{I}}^1,1)$-criticality can be checked in linear time for trees.

15 pages, minor presentation improvements made