Super Edge-Connectedness Keeping Tree

• Super edge-connectedness keeping trees are subtrees that, when removed, preserve a graph's strongest form of connectivity by ensuring every minimum edge-cut isolates only one vertex.
• They play a key role in graph decomposition, particularly in cographs, where cotree and tree-cut frameworks efficiently characterize connectivity-preserving structures.
• Their existence relies on stringent degree conditions and is validated by tight extremal examples and polynomial-time algorithms leveraging max-flow and combinatorial techniques.

A super edge-connectedness keeping tree is a subtree whose removal preserves the strongest form of edge-connectivity, termed super edge-connectivity, in the host graph. The concept arises at the intersection of edge-connectivity theory, canonical tree-like graph decompositions, and extremal combinatorics, and has been distinctly characterized for cographs—a class of graphs admitting a cotree decomposition and characterized as $P_4$-free. This notion is situated at the apex of a hierarchy of "connectivity-keeping" tree concepts, generalizing earlier frameworks for vertex- and edge-connectivity preservation.

1. Definition and Fundamental Concepts

A connected graph $G=(V,E)$ is super edge-connected if every minimum edge-cut isolates a single vertex. That is, for all edge sets $F \subseteq E(G)$ with $|F| = \lambda(G)$ (where $\lambda(G)$ is the edge-connectivity), the subgraph $G - F$ is disconnected only if $F$ removes all edges incident to some vertex $v$, i.e., $F = \{vw \mid w \in N_G(v)\}$ for some $v$ with $d_G(v) = \delta(G)$, the minimum degree. This definition is extended to include disconnected graphs with exactly one isolated vertex as super edge-connected.

A super edge-connectedness keeping tree in $G$ with respect to a given tree $T$ (of order $m$) is a subtree $T' \subseteq G$ with $T' \cong T$ such that $G - V(T')$ is again super edge-connected (Hasunuma, 16 Nov 2025).

For general graphs (finite or infinite), a canonical hierarchical decomposition exists into $k$-edge-connected pieces for all $k \in \mathbb{N} \cup \{\infty\}$ simultaneously, via a tree-cut decomposition and a nested set of bonds. The tree structure efficiently encodes all possible edge-block decompositions for varying $k$, culminating in the super edge-connectedness keeping tree for the highest level of connectivity (Elbracht et al., 2020).

2. Existence Theorems in Cographs

For cographs, the existence and tightness of super edge-connectedness keeping trees is established as follows: Let $G$ be a super edge-connected cograph and $T$ any tree of order $m$. If $\delta(G) \geq m + 2$, then there exists a subtree $T' \cong T$ such that $G - V(T')$ is super edge-connected. This degree bound is best possible; the construction

$$H_i = (K_1 \cup K_1) + (K_1 \cup K_1) + \cdots + (K_1 \cup K_1),\quad G = (H_1 \cup H_2) + K_1$$

shows that $\delta(G) = m+1$ is insufficient when $T$ is a star $K_{1,m-1}$ (Hasunuma, 16 Nov 2025).

3. Decomposition and Construction Principles

For any connected graph, a canonical "nested" family $N(G)$ of bonds is constructed such that, for each $k \in \mathbb{N}$, the subfamily $N_k := \{F \in N(G) : |F| < k\}$ yields the set of fundamental cuts of a tree–cut decomposition $(T_k,(X_t))$. In this decomposition, each node $t$ of $T_k$ corresponds bijectively to a maximal $k$-edge-connected piece (the $k$-edge-block), and every edge of $G$ is either absorbed in one block or becomes an edge of $T_k$ by crossing a unique fundamental cut (Elbracht et al., 2020).

The construction relies on a combinatorial approach, utilizing the family of efficient minimal bonds separating pairs of edge-blocks. The nested set theorem ("thinly splinters" lemma) guarantees a unique canonical nested family meeting every such separator. For each $k$, absence of separators of order $<k$ ensures the preservation of the corresponding edge-connectivity upon deletion of subtrees.

4. Characterization in Cographs

For cographs, super edge-connectivity is fully characterized: a connected cograph is always maximally edge-connected, and is super edge-connected if and only if (i) it is not $C_4$, and (ii) it is not of the form $H+K_1$ with $H$ a disconnected cograph of order $\geq 4$ containing a component isomorphic to $K_{\delta(G)}$. Cotree decomposition underpins the partitioning into cocomponents. The proof of the main theorem bifurcates according to the size of $S = V(G) \setminus V(G_1)$, where $G_1$ is the primary cocomponent: if $|S| \geq 2$ the result follows by 2-connectivity arguments, otherwise $G=G_1+K_1$ and the construction operates within $G_1$ (Hasunuma, 16 Nov 2025).

5. Hierarchy of Connectivity-Keeping Trees

The super edge-connectedness keeping tree is the most stringent member in a hierarchy of connectivity-keeping trees in cographs. Each stronger notion demands a higher minimum-degree condition:

Notion	Minimum Degree Condition
Vertex-connectivity keeping tree	$\delta(G)\ge\lfloor 3k/2\rfloor+m-1$
Edge-connectivity keeping tree	$\delta(G)\ge k+m-[k=1]$
Super edge-connectivity keeping tree	$\delta(G)\ge m+2$

This ordering reflects the structural strengthening from vertex- and edge-connectivity to super edge-connectivity, with the latter ensuring that all minimum cuts isolate only single vertices. The result for super edge-connectedness keeping trees marks the culmination of this hierarchy, both in satisfying the largest minimum-degree hypothesis and enforcing the strongest preservation criteria (Hasunuma, 16 Nov 2025).

6. Algorithmic and Structural Connections

In finite graphs, the construction of canonical nested families $N(G)$ can be realized by repeated max-flow computations between block pairs, yielding polynomial time complexity. For $k = \infty$, the resulting tree recovers the classical Gomory–Hu tree, and for $k=3$ the construction generalizes Tutte's decomposition of 2-connected graphs into 3-connected components. In infinite cases, Menger-type arguments and ray–comb techniques ensure the correct infinite behavior of decomposition trees (Elbracht et al., 2020).

For cographs, repeated application of cotree decomposition, tree-extension lemmas, and careful handling of component structure facilitate the efficient construction of super edge-connectedness keeping trees.

7. Extremal and Illustrative Examples

Extremal constructions show that the degree bound for the existence of a super edge-connectedness keeping tree in cographs is tight. For even $m$,

$$G = \left( \bigcup_{i=1}^2 H_i \right) + K_1 \text{ with } H_i = \text{join of } m/2+1 \text{ copies of } K_1 \cup K_1$$

achieves $\delta(G) = m+1$ but fails to admit a super edge-connectedness keeping tree for the star $K_{1,m-1}$, while $\delta(G) = m+2$ suffices for all trees of order $m$ (Hasunuma, 16 Nov 2025).

A simple finite example illustrates the tree-cut approach: a graph formed by two triangles connected by two parallel edges has, for $k=2$, the two triangles as 2-edge-blocks separated by a bond of size 2, and for $k=3$ the entire graph as a single 3-edge-block, in line with the canonical decomposition framework (Elbracht et al., 2020).