Isolation critical graphs under multiple edge subdivision

Abstract

This paper introduces the notion of $(ι,q)$-critical graphs. The isolation number of a graph $G$, denoted by $ι(G)$ and also known as the vertex-edge domination number, is the minimum number of vertices in a set $D$ such that the subgraph induced by the vertices not in the closed neighbourhood of $D$ has no edges. A graph $G$ is $(ι,q)$-critical, $q \ge 1$, if the subdivision of any $q$ edges in $G$ gives a graph with isolation number greater than $ι(G)$ and there exists a set of $q-1$ edges such that subdividing them gives a graph with isolation number equal to $ι(G)$. We prove that for each integer $q \ge 1$ there exists a $(ι,q)$-critical graph, while for a given graph $G$, the admissible values of $q$ satisfy $1 \le q \le |E(G)| - 1$. In addition, we provide a general characterisation of $(ι,1)$-critical graphs as well as a constructive characterisation of $(ι,1)$-critical trees.

Figures (2)

• Figure 1: The graph $Q_3$ with the edges of $A_3$ coloured red. Note that $\iota(Q_3) = \iota({(Q_3)}_{A_3})$.
• Figure 2: Operations used in the constructive characterization of $(\iota,1)$-critical trees.

Theorems & Definitions (24)

• Proposition 2.1
• Corollary 2.2
• Proposition 3.1
• proof
• Corollary 3.2
• Lemma 3.3: Borg1
• Lemma 3.4: Borg1Borgrsc
• Lemma 3.5
• Theorem 3.6
• Corollary 3.7