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 $G=(V,E)$0, 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=(V,E)$1 with respect to a given tree $G=(V,E)$2 (of order $G=(V,E)$3) is a subtree $G=(V,E)$4 with $G=(V,E)$5 such that $G=(V,E)$6 is again super edge-connected (Hasunuma, 16 Nov 2025).

For general graphs (finite or infinite), a canonical hierarchical decomposition exists into $G=(V,E)$7-edge-connected pieces for all $G=(V,E)$8 simultaneously, via a tree-cut decomposition and a nested set of bonds. The tree structure efficiently encodes all possible edge-block decompositions for varying $G=(V,E)$9, 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 $F \subseteq E(G)$0 be a super edge-connected cograph and $F \subseteq E(G)$1 any tree of order $F \subseteq E(G)$2. If $F \subseteq E(G)$3, then there exists a subtree $F \subseteq E(G)$4 such that $F \subseteq E(G)$5 is super edge-connected. This degree bound is best possible; the construction

$F \subseteq E(G)$6

shows that $F \subseteq E(G)$7 is insufficient when $F \subseteq E(G)$8 is a star $F \subseteq E(G)$9 (Hasunuma, 16 Nov 2025).

3. Decomposition and Construction Principles

For any connected graph, a canonical "nested" family $|F| = \lambda(G)$0 of bonds is constructed such that, for each $|F| = \lambda(G)$1, the subfamily $|F| = \lambda(G)$2 yields the set of fundamental cuts of a tree–cut decomposition $|F| = \lambda(G)$3. In this decomposition, each node $|F| = \lambda(G)$4 of $|F| = \lambda(G)$5 corresponds bijectively to a maximal $|F| = \lambda(G)$6-edge-connected piece (the $|F| = \lambda(G)$7-edge-block), and every edge of $|F| = \lambda(G)$8 is either absorbed in one block or becomes an edge of $|F| = \lambda(G)$9 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 $\lambda(G)$0, absence of separators of order $\lambda(G)$1 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 $\lambda(G)$2, and (ii) it is not of the form $\lambda(G)$3 with $\lambda(G)$4 a disconnected cograph of order $\lambda(G)$5 containing a component isomorphic to $\lambda(G)$6. Cotree decomposition underpins the partitioning into cocomponents. The proof of the main theorem bifurcates according to the size of $\lambda(G)$7, where $\lambda(G)$8 is the primary cocomponent: if $\lambda(G)$9 the result follows by 2-connectivity arguments, otherwise $G - F$0 and the construction operates within $G - F$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	$G - F$2
Edge-connectivity keeping tree	$G - F$3
Super edge-connectivity keeping tree	$G - F$4

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 $G - F$5 can be realized by repeated max-flow computations between block pairs, yielding polynomial time complexity. For $G - F$6, the resulting tree recovers the classical Gomory–Hu tree, and for $G - F$7 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 $G - F$8,

$G - F$9

achieves $F$0 but fails to admit a super edge-connectedness keeping tree for the star $F$1, while $F$2 suffices for all trees of order $F$3 (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 $F$4, the two triangles as 2-edge-blocks separated by a bond of size 2, and for $F$5 the entire graph as a single 3-edge-block, in line with the canonical decomposition framework (Elbracht et al., 2020).