Solution to a $3$-path isolation problem for subcubic graphs

TL;DR

The paper resolves the asymptotic $P_3$-isolation problem for connected subcubic graphs with no induced $6$-cycles by proving $ι(G,P_3)\le n/4$, with equality only for the Borg construction $B_{n,P_3}$ and exceptions comprising 12 explicit graphs. It introduces the key extremal framework, including the $B_{n,F}$ construction and the $\mathcal{E}$-graphs (the $G_{3}, G_{7,i}, G_{11}, G_{15}$ family), which are the precise graphs that violate the bound. The main method combines induction on $n$ with a component-decomposition around a degree-$3$ vertex, classifies components as $\mathcal{E}$-graphs or not, and builds a $P_3$-isolating set by aggregating component-wise isolating sets with a tightly controlled neighborhood around $N[v]$, using extensive case analysis. This work isolates the impact of induced $6$-cycles in the subcubic setting and yields a tight, structural characterization of extremal graphs for $P_3$-isolation.

Abstract

The $3$-path isolation number of a connected $n$-vertex graph $G$, denoted by $ι(G,P_3)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$ intersects each $3$-vertex path of $G$, meaning that no two edges of $G-N[D]$ intersect. If $G$ is not a $3$-path or a $3$-cycle or a $6$-cycle, then $ι(G,P_3) \leq 2n/7$. This was proved by Zhang and Wu, and independently by Borg in a slightly extended form. The bound is attained by infinitely many graphs having induced $6$-cycles. Huang, Zhang and Jin showed that if $G$ has no $6$-cycles, or $G$ has no induced $5$-cycles and no induced $6$-cycles, then $ι(G,P_3) \leq n/4$ unless $G$ is a $3$-path or a $3$-cycle or a $7$-cycle or an $11$-cycle. They asked if the bound still holds asymptotically for connected graphs having no induced $6$-cycles. Thus, the problem essentially is whether induced $6$-cycles solely account for the difference between the two bounds. In this paper, we solve this problem for subcubic graphs, which need to be treated differently from other graphs. We show that if $G$ is subcubic and has no induced $6$-cycles, then $ι(G,P_3) \leq n/4$ unless $G$ is a copy of one of $12$ particular graphs whose orders are $3$, $7$, $11$ and $15$. The bound is sharp.

Figures (3)

• Figure 2: $G_{11}$, a subcubic $11$-vertex graph with no induced 6-cycles and with $P_3$-isolation number $3$.
• Figure 3: $G_{15}$, a subcubic $15$-vertex graph with no induced $6$-cycles and with $P_3$-isolation number $4$.
• Figure a: $G_{7,1}$

Theorems & Definitions (11)

• Theorem 1
• Lemma 1: Borg1
• Lemma 2: Borg1
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9