Isolation critical graphs under multiple edge subdivision

TL;DR

This paper introduces and characterizes $(ta,q)$-critical graphs based on the isolation number, showing their existence for all $q \u2265 1$, and provides bounds and efficient recognition methods for trees.

Contribution

It defines the concept of $(ta,q)$-critical graphs, proves their existence for all $q \u2265 1$, and characterizes them, especially for trees, with linear-time recognition.

Findings

$(ta,q)$-critical graphs exist for all $q \u2265 1$

Connected non-star graphs are $(ta,q)$-critical for some $q \u2264 m-1$

Linear-time recognition for $(ta,1)$-critical trees

Abstract

This paper introduces the notion of an $(\iota,q)$-critical graph. The isolation number of a graph $G$, denoted by $\iota(G)$ 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 $(\iota,q)$-critical if every subdivision of $q$ edges of $G$ gives a graph whose isolation number is greater than $\iota(G)$, and $G$ has $q-1$ edges such that subdividing them gives a graph whose isolation number is $\iota(G)$. We show that an $(\iota,q)$-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 $(\iota,q)$-critical for some $q \le m - 1$. We show that this bound is best possible. We provide a general characterization of $(\iota,1)$-critical graphs as well as a constructive characterization of $(\iota,1)$-critical trees, demonstrating that $(\iota,1)$-criticality can be checked in linear time for trees.