Bounds on Independent Isolation in Graphs

TL;DR

This work studies the independent isolation number $\iota^i(G)$, the minimum size of an independent isolating set in a graph, as a variation of isolation concepts restricted to independence. It establishes a tight $n/3$ bound for connected bipartite graphs by a three-way partition and proves a $(n+1)/3$ upper bound for connected $3$-colorable graphs using a rotation-sweep argument on a $3$-coloring, while showing the bound is best possible via constructive examples and a boosting operation. The paper also derives bounds for cubic and maximal outerplanar graphs, and extends the study to $k$-colorable graphs, proving a linear bound $\iota^i(G) \le c_k n$ with an explicit bound $\iota^i(G) \le (k+2)n/(2k+6)$. Additionally, it proves that two disjoint independent isolating sets always exist, but three such sets is NP-complete to decide, via a gadget construction that links 3-colored structure to a 4-coloring problem. Overall, the results illuminate how colorability and graph structure constrain independent isolation and point to rich directions for future exploration of $\mathcal{F}$-isolation with independence constraints.

Abstract

An isolating set of a graph is a set of vertices $S$ such that, if $S$ and its neighborhood is removed, only isolated vertices remain; and the isolation number is the minimum size of such a set. It is known that for every connected graph apart from $K_2$ and $C_5$, the isolation number is at most one-third the order and indeed such a graph has three disjoint isolating sets. In this paper we consider isolating sets where $S$ is required to be an independent set and call the minimum size thereof the independent isolation number. While for general graphs of order $n$ the independent isolation number can be arbitrarily close to $n/2$, we show that in bipartite graphs the vertex set can be partitioned into three disjoint independent isolating sets, whence the independent isolation number is at most $n/3$; while for $3$-colorable graphs the maximum value of the independent isolation number is $(n+1)/3$. We also provide a bound for $k$-colorable graphs.

Figures (7)

• Figure 1: The graph $M_4$
• Figure 2: Construction of $J(G)$
• Figure 3: The $P_2$-corona of $C_6$
• Figure 4: The construction of $D_e$ (shaded) and $F_e$ for $e=vw$
• Figure 5: A $3$-colorable graph with $n=17$ and $\iota^i =6$

Theorems & Definitions (15)

• Lemma 1
• Lemma 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10