Super-minimally $3$-connected graphs

TL;DR

The paper investigates the structure of super-minimally $3$-connected graphs, defining this class and situating it between minimally and uniformly $3$-connected graphs. It develops a hierarchy of connectivity notions and uses a combination of contraction- and decomposition-based arguments along with the bridging, enhanced deletion, and cleaving operations to derive sharp extremal bounds. It proves that such graphs have at least $\frac{|V(G)|+3}{2}$ degree-$3$ vertices and at most $|E(G)|\le 2|V(G)|-2$, with equality precisely for wheels with at least three spokes, and constructs infinite families attaining these bounds (e.g., belts $B_n$). These results extend Halin's and Xu's prior work and sharpen the understanding of the relationship between different connectivity notions, offering explicit extremal examples and a toolkit for structural analysis.

Abstract

In this paper, we introduce super-minimally $k$-connected graphs, those $k$-connected graphs in which no proper subgraph is $k$-connected. For $k$ greater than or equal to three, this class lies strictly between the classes of minimally $k$-connected graphs and uniformly $k$-connected graphs. In particular, we determine the minimum number of degree-$3$ vertices in a super-minimally $3$-connected graph, thereby extending a result of Halin on minimally $3$-connected graphs. In addition, we determine the maximum number of edges in a super-minimally $3$-connected graph, extending Xu's result for uniformly $3$-connected graphs, and providing an analogue of Halin's result for minimally $3$-connected graphs.

Figures (19)

• Figure 1: $\{G_1,G_2\}$ is a $2$-separation of $G.$
• Figure 2: (a). A $4$-dimensional wheel $W(3,3,3,3)$, and (b). a $3$-connected proper subgraph.
• Figure 3: An alternating double wheel $A_4$.
• Figure 4: $Q_5$ is super-minimally $4$-connected, but not uniformly $4$-connected.
• Figure 5: A Venn diagram of super-minimally, minimally, critically, and uniformly $3$-connected graphs.

Theorems & Definitions (47)

• Lemma 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 1.6
• Theorem 1.7
• Conjecture 1
• Theorem 1.8
• Theorem 1.9