Mader Conjecture in Tree Embedding

• Mader Conjecture is a graph-theoretic proposition stating that every k-connected graph with δ(G) ≥ ⌊3k/2⌋+m−1 contains a subtree isomorphic to any given tree of order m while preserving connectivity after removal.
• Historical results confirm the conjecture for k=1, 2, and 3, with proofs addressing paths, stars, and double-stars through layered embedding and maximality arguments.
• Recent studies extend confirmations to cographs and specialized tree families, yet the full conjecture remains open for k ≥ 4, inviting novel combinatorial and structural techniques.

Searching arXiv for recent and foundational papers on Mader's conjecture in the connectivity-preserving tree-embedding setting. Mader’s conjecture, in the graph-theoretic sense addressed here, is a degree condition for embedding a prescribed tree in a highly connected graph while preserving the original connectivity after the embedded tree is deleted. Formally, for positive integers $k,m$ and a tree $T$ of order $m$, the conjecture asserts that every $k$-connected graph $G$ with

$$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$

contains a subtree $T'\subseteq G$ with $T'\cong T$ such that $G-V(T')$ is still $k$-connected (Hong et al., 2021). The conjecture was proposed by Mader as a strengthening of earlier connectivity-preserving embedding problems, and the subsequent literature has developed both exact confirmations in low-connectivity cases and partial results for restricted tree classes and graph classes (Ji et al., 2020, Tian et al., 2017, Hasunuma, 16 Nov 2025).

1. Formal statement and basic notions

A graph $T$0 is $T$1-connected if it has more than $T$2 vertices and remains connected after the deletion of any set of fewer than $T$3 vertices. Its minimum degree is denoted by $T$4. A tree $T$5 of order $T$6 means $T$7. A subtree $T$8 is isomorphic to $T$9, written $m$0, if there is a bijection $m$1 preserving adjacency (Hong et al., 2021).

In this notation, Mader’s conjecture states that for every positive integers $m$2 and every tree $m$3 with $m$4, every $m$5-connected graph $m$6 satisfying

$m$7

contains a subtree $m$8 with $m$9 such that $k$0 is still $k$1-connected (Hong et al., 2021). In equivalent language used elsewhere in the literature, this requires

$k$2

for the chosen copy $k$3 (Ji et al., 2020).

The conjecture is often described as a non-separating tree problem: the embedded tree must be present as a subgraph, but its vertex set must not destroy the ambient graph’s $k$4-connectivity when removed. This places the conjecture at the interface of tree embedding, connectivity, fragment structure, and extremal degree conditions.

2. Historical development and verified cases

Several exact cases are known. The literature summarized here states that Diwan–Tholiya settled the conjecture for $k$5 (Tian et al., 2017), and that Mader proved the conjecture when $k$6 is a path (Ji et al., 2020). A further milestone was the confirmation for $k$7 in full generality for arbitrary trees (Hong et al., 2021).

The paper "Mader's conjecture for graphs with small connectivity" (Hong et al., 2021) establishes the conjecture for $k$8. In particular, it proves:

$k$9

and

$G$0

These are Theorem B and Theorem C in the paper (Hong et al., 2021).

Before the full $G$1 result was obtained, several special tree families had been handled at the conjectured threshold. For 2-connected graphs, stars and double-stars were proved by Tian and collaborators under the degree condition $G$2 (Tian et al., 2017). Further work extended the $G$3 theory to path-star and path-double-star families at the exact bound $G$4 (Tian et al., 2017). These partial results played a structural role by showing that branching trees beyond paths could be embedded non-separatingly under the predicted degree-order bound.

For $G$5, the conjecture remains open in general (Hong et al., 2021, Ji et al., 2020).

3. Structural methods in the low-connectivity proofs

The low-connectivity breakthrough in (Hong et al., 2021) rests on a general embedding characterization and two different maximality schemes for the cases $G$6 and $G$7.

A central ingredient is an embedding-with-reserved-vertex statement. Let $G$8 be a tree of order $G$9 and $$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$0 a graph with

$$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$1

Suppose $$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$2 are disjoint subgraphs of $$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$3 with $$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$4 and

$$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$5

Then one of the following holds:

1. $$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$6;
2. for every $$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$7,

$$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$8

1. there exists $$\delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1$$9 with

$T'\subseteq G$0

and $T'\subseteq G$1 contains a copy of $T'\subseteq G$2 (Hong et al., 2021).

This characterization is the mechanism that permits re-embedding the designated tree while avoiding a strategically chosen vertex. The paper derives it from a layered embedding lemma. In that lemma, the vertices still to be embedded are partitioned into layers

$T'\subseteq G$3

and the available vertices of the host graph are split as

$T'\subseteq G$4

with the inductive condition

$T'\subseteq G$5

ensuring that the embedding can be extended greedily from a core subtree outward (Hong et al., 2021). Corollary 2.4 specializes this to three layers: $T'\subseteq G$6

For $T'\subseteq G$7, the proof chooses a maximal subtree $T'\subseteq G$8 so that a largest block $T'\subseteq G$9 in $T'\cong T$0 is as large as possible. Applying the embedding characterization to the remainder $T'\cong T$1 yields a contradiction unless $T'\cong T$2, which implies that $T'\cong T$3 is 2-connected (Hong et al., 2021).

For $T'\cong T$4, the block argument is replaced by a more rigid object: an induced subgraph $T'\cong T$5 which is a subdivision of some simple 3-connected graph, with $T'\cong T$6, the number of vertices of degree at least $T'\cong T$7, chosen maximal. Ear-decomposition arguments and four successive claims then show that any leftover piece $T'\cong T$8 either permits re-embedding of the tree or forces a strictly larger subdivision in $T'\cong T$9, contradicting maximality (Hong et al., 2021). This use of subdivisions of 3-connected graphs marks a distinct increase in structural complexity from the $G-V(T')$0 case.

4. Restricted tree classes and specialized confirmations

A substantial part of the literature consists of confirmations for special tree families, especially when $G-V(T')$1 or when additional hypotheses are imposed on the host graph.

For stars and double-stars in 2-connected graphs, (Tian et al., 2017) proves that if $G-V(T')$2 is 2-connected with $G-V(T')$3, then every star $G-V(T')$4 of order $G-V(T')$5 and every double-star $G-V(T')$6 of order $G-V(T')$7 has a copy $G-V(T')$8 such that $G-V(T')$9. A key auxiliary statement is an explicit embedding lemma for a double-star obtained from an edge $k$0 by attaching $k$1 leaves to $k$2 and $k$3 leaves to $k$4, with $k$5. If an edge $k$6 satisfies

$k$7

then $k$8 contains a copy of the prescribed double-star centered on $k$9 (Tian et al., 2017). The proofs then proceed by choosing an extremal copy $T$00, analyzing the maximum block $T$01 of $T$02, and repeatedly re-embedding the tree to contradict maximality.

The paper (Tian et al., 2017) extends the $T$03 theory to two additional infinite classes. A path-star $T$04 is obtained by identifying one end of a path of order $T$05 with one leaf of a star of order $T$06. A path-double-star is defined analogously from a path and a double-star. The paper proves that every 2-connected graph $T$07 with

$T$08

contains such a tree $T$09 as a subgraph with $T$10 still 2-connected (Tian et al., 2017). The underlying method combines Hamidoune’s fragment lemma with a cut-preserving lemma attributed to Mader.

For spiders, (Ji et al., 2020) proves a different type of restricted result. A spider is a tree with at most one vertex of degree at least $T$11; if its leg-lengths are $T$12, it is denoted by

$T$13

The main theorem states that if $T$14 is $T$15-connected and satisfies

$T$16

then for every spider $T$17 of order $T$18, the graph $T$19 contains a copy $T$20 such that

$T$21

(Ji et al., 2020). The proof uses fragment machinery, end-fragments, and induction on $T$22, together with a maximal spider that is extended by analyzing a suitable end-fragment of the remainder.

These results do not settle the full conjecture for higher $T$23, but they isolate mechanisms that operate effectively for low branching complexity or for nearly complete host graphs.

5. Class-restricted results: cographs

A more recent development concerns the conjecture on cographs, equivalently $T$24-free graphs. The paper "Mader's Conjecture and Its Variants for Cographs" (Hasunuma, 16 Nov 2025) proves that for any tree $T$25 of order $T$26, every $T$27-connected cograph $T$28 with

$T$29

contains a subtree $T$30 such that $T$31. Thus the conjectured degree bound is valid throughout the cograph class (Hasunuma, 16 Nov 2025).

The proof exploits the cotree decomposition of cographs. Any nontrivial connected cograph can be written as

$T$32

the join of its cocomponents, with $T$33. A decisive structural identity is

$T$34

where $T$35 (Hasunuma, 16 Nov 2025). The authors also use a $T$36-keeping lemma asserting that if $T$37 is $T$38-connected, $T$39 with $T$40, and $T$41 with $T$42, then the induced subgraph on $T$43 is $T$44-connected (Hasunuma, 16 Nov 2025).

The cograph paper also establishes three variants of the conjecture for cographs, including edge-deletion and edge-connectivity versions. For example, every $T$45-connected cograph $T$46 with

$T$47

contains a subtree $T$48 such that $T$49 is still $T$50-connected, and every $T$51-edge-connected cograph with

$T$52

contains $T$53 such that $T$54 is $T$55-edge-connected (Hasunuma, 16 Nov 2025). These class-specific results show that the conjectured threshold is compatible with highly decomposable graph classes in which connectivity can be expressed explicitly.

6. Open problems and broader context

For general graphs, the conjecture is completely confirmed only for $T$56 (Hong et al., 2021). The main open range is $T$57. The literature emphasizes that the best known universal bound, proved by Mader in 2012, is

$T$58

which is far above the conjectured

$T$59

(Hong et al., 2021, Ji et al., 2020). This gap quantifies the remaining difficulty.

The 2021 low-connectivity paper explicitly raises the question of whether the characterization underlying its 2-connected proof can be generalized to $T$60: can one extend the embedding lemma with reserved vertices and still force the required non-separating embedding under the conjectured degree bound (Hong et al., 2021)? The same paper suggests that the layered-embedding and ear-decomposition approach may extend to higher $T$61, but that intricate multiple-ear interactions pose new challenges (Hong et al., 2021).

A related obstacle, already visible in earlier partial results, is the combinatorial proliferation of ways in which a subtree can interact with a $T$62-separator. One source states that for general $T$63, one would need to control all $T$64 ways that a subtree can meet a $T$65-separator (Tian et al., 2017). This suggests that the conjecture becomes increasingly sensitive to separator geometry as $T$66 grows.

Within the current body of results, three tendencies are clear. First, exact proofs at the conjectured threshold are available for low $T$67 and for several nontrivial tree families. Second, structural graph classes such as cographs admit full confirmations via decomposition formulas rather than separator-maximality arguments. Third, higher-connectivity cases appear to require new methods capable of simultaneously managing embeddings, separators, and large-scale connectivity-preserving reconfiguration (Hong et al., 2021, Hasunuma, 16 Nov 2025).

Taken together, these results position Mader’s conjecture as a central problem in connectivity-preserving subgraph embedding: it is exact in formulation, sharp in the known cases, and still unresolved precisely where separator complexity and global structure begin to dominate the local degree condition.