--- title: Mader Conjecture in Tree Embedding url: https://www.emergentmind.com/topics/mader-conjecture type: topic --- # Mader Conjecture in Tree Embedding 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 [2101.11777]. 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 [2012.04816], [1707.01165], [2511.12499]. ## 1. Formal statement and basic notions A graph \(G\) is \(k\)-connected if it has more than \(k\) vertices and remains connected after the deletion of any set of fewer than \(k\) vertices. Its minimum degree is denoted by \(\delta(G)\). A tree \(T\) of order \(m\) means \(|V(T)|=m\). A subtree \(T'\subseteq G\) is isomorphic to \(T\), written \(T'\cong T\), if there is a bijection \(\varphi:V(T)\to V(T')\) preserving adjacency [2101.11777]. In this notation, Mader’s conjecture states that for every positive integers \(k,m\) and every tree \(T\) with \(|V(T)|=m\), every \(k\)-connected graph \(G\) satisfying \[ \delta(G)\;\ge\;\Big\lfloor \tfrac{3k}{2}\Big\rfloor \;+\;m\;-\;1 \] contains a subtree \(T'\subseteq G\) with \(T'\cong T\) such that \(G - V(T')\) is still \(k\)-connected [2101.11777]. In equivalent language used elsewhere in the literature, this requires \[ \kappa\bigl(G - V(T')\bigr)\;\ge\;k \] for the chosen copy \(T'\) [2012.04816]. 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 \(k\)-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 \(k=1\) [1707.01165], and that Mader proved the conjecture when \(T\) is a path [2012.04816]. A further milestone was the confirmation for \(k=2,3\) in full generality for arbitrary trees [2101.11777]. The paper "Mader's conjecture for graphs with small connectivity" [2101.11777] establishes the conjecture for \(k=1,2,3\). In particular, it proves: \[ \text{If }G\text{ is 2-connected and }\delta(G)\ge m+2, \text{ then }G\text{ contains }T'\cong T_0\text{ with }G-V(T')\text{ 2-connected;} \] and \[ \text{if }G\text{ is 3-connected and }\delta(G)\ge m+3, \text{ then }G\text{ contains }T'\cong T_0\text{ with }G-V(T')\text{ 3-connected.} \] These are Theorem B and Theorem C in the paper [2101.11777]. Before the full \(k=2\) 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 \(\delta(G)>m+2\) [1707.01165]. Further work extended the \(k=2\) theory to path-star and path-double-star families at the exact bound \(\delta(G)\ge m+2\) [1710.01883]. 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 \(k\ge 4\), the conjecture remains open in general [2101.11777], [2012.04816]. ## 3. Structural methods in the low-connectivity proofs The low-connectivity breakthrough in [2101.11777] rests on a general embedding characterization and two different maximality schemes for the cases \(k=2\) and \(k=3\). A central ingredient is an embedding-with-reserved-vertex statement. Let \(T_0\) be a tree of order \(m\) and \(G\) a graph with \[ \delta(G)\;\ge\;m+2. \] Suppose \(T,B,H\) are disjoint subgraphs of \(G\) with \(T\cong T_0\) and \[ V(G)\;=\;V(H)\,\dot\cup\,V(B)\,\dot\cup\,V(T). \] Then one of the following holds: 1. \(N_G(H)\cap V(T)=\emptyset\); 2. for every \(v\in V(H\cup T)\), \[ \bigl|N_G(v)\cap V(B)\bigr|\le\delta(G)-m; \] 3. there exists \(v\in V(H\cup T)\) with \[ \bigl|N_G(v)\cap V(B)\bigr|\ge\delta(G)-m+1 \] and \((H\cup T)-v\) contains a copy of \(T_0\) [2101.11777]. 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 \[ (L_1,\dots,L_{k+1}) \] and the available vertices of the host graph are split as \[ (X_1,\dots,X_{k+1}), \] with the inductive condition \[ |N_G(x)\cap \bigcup_{jm+2\), then every star \(T\) of order \(m\) and every double-star \(T\) of order \(m\ge 5\) has a copy \(T'\cong T\) such that \(\kappa(G-V(T'))\ge 2\). A key auxiliary statement is an explicit embedding lemma for a double-star obtained from an edge \(u'v'\) by attaching \(r\) leaves to \(u'\) and \(s\) leaves to \(v'\), with \(r+s=m-2\). If an edge \(uv\in E(G)\) satisfies \[ \lvert N_G(u)\setminus\{v\}\rvert\ge r,\quad \lvert N_G(v)\setminus\{u\}\rvert\ge s,\quad \bigl\lvert(N_G(u)\cup N_G(v))\setminus\{u,v\}\bigr\rvert\ge m-2, \] then \(G\) contains a copy of the prescribed double-star centered on \(uv\) [1707.01165]. The proofs then proceed by choosing an extremal copy \(T'\), analyzing the maximum block \(B\) of \(G-V(T')\), and repeatedly re-embedding the tree to contradict maximality. The paper [1710.01883] extends the \(k=2\) theory to two additional infinite classes. A path-star \(PS(r,m-r)\) is obtained by identifying one end of a path of order \(r+1\) with one leaf of a star of order \(m-r\). A path-double-star is defined analogously from a path and a double-star. The paper proves that every 2-connected graph \(G\) with \[ \delta(G)\ge m+2 \] contains such a tree \(T\) as a subgraph with \(G-V(T)\) still 2-connected [1710.01883]. The underlying method combines Hamidoune’s fragment lemma with a cut-preserving lemma attributed to Mader. For spiders, [2012.04816] proves a different type of restricted result. A spider is a tree with at most one vertex of degree at least \(3\); if its leg-lengths are \(t_1,\dots,t_t\), it is denoted by \[ T_{t_1,t_2,\dots,t_t}. \] The main theorem states that if \(G\) is \((k+1)\)-connected and satisfies \[ \delta(G)\;\ge\;\Big\lfloor\tfrac{3k}{2}\Big\rfloor + m - 1 \quad\text{and}\quad \delta(G)=|V(G)|-1, \] then for every spider \(T_{t\,;\;m-t-1}\) of order \(m\), the graph \(G\) contains a copy \(T'\) such that \[ \kappa\bigl(G - V(T')\bigr)\;\ge\;k \] [2012.04816]. The proof uses fragment machinery, end-fragments, and induction on \(|V(G)|\), 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 \(k\), 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 \(P_4\)-free graphs. The paper "Mader's Conjecture and Its Variants for Cographs" [2511.12499] proves that for any tree \(T\) of order \(m\), every \(k\)-connected cograph \(G\) with \[ \delta(G)\;\ge\;\Big\lfloor\frac{3k}{2}\Big\rfloor + m -1 \] contains a subtree \(T'\cong T\) such that \(\kappa(G-V(T'))\ge k\). Thus the conjectured degree bound is valid throughout the cograph class [2511.12499]. The proof exploits the cotree decomposition of cographs. Any nontrivial connected cograph can be written as \[ G \;=\; G_1 + G_2 + \cdots + G_{t_G}, \] the join of its cocomponents, with \(|V(G_1)|\ge |V(G_2)|\ge \cdots\). A decisive structural identity is \[ \kappa(G)=n_G-n'_G, \] where \(n'_G=|V(G_1)|\) [2511.12499]. The authors also use a \(k\)-keeping lemma asserting that if \(G=G_1+\cdots+G_{t_G}\) is \(k\)-connected, \(S_1\subseteq V(G_1)\) with \(|S_1|\ge \min\{k,|V(G_1)|\}\), and \(S_2\subseteq V(G)\setminus V(G_1)\) with \(|S_2|\ge k\), then the induced subgraph on \(S_1\cup S_2\) is \(k\)-connected [2511.12499]. The cograph paper also establishes three variants of the conjecture for cographs, including edge-deletion and edge-connectivity versions. For example, every \(k\)-connected cograph \(G\) with \[ \delta(G)\;\ge\;k+m-1 \] contains a subtree \(T'\cong T\) such that \(G-E(T')\) is still \(k\)-connected, and every \(k\)-edge-connected cograph with \[ \delta(G)\;\ge\;k+m-[k=1] \] contains \(T'\cong T\) such that \(G-V(T')\) is \(k\)-edge-connected [2511.12499]. 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 \(k\le 3\) [2101.11777]. The main open range is \(k\ge 4\). The literature emphasizes that the best known universal bound, proved by Mader in 2012, is \[ \delta(G)\;\ge\;2\bigl(k-1+m\bigr)^2 \;+\;m\;-\;1, \] which is far above the conjectured \[ \Big\lfloor \tfrac{3k}{2}\Big\rfloor + m - 1 \] [2101.11777], [2012.04816]. 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 \(k\ge 4\): can one extend the embedding lemma with reserved vertices and still force the required non-separating embedding under the conjectured degree bound [2101.11777]? The same paper suggests that the layered-embedding and ear-decomposition approach may extend to higher \(k\), but that intricate multiple-ear interactions pose new challenges [2101.11777]. A related obstacle, already visible in earlier partial results, is the combinatorial proliferation of ways in which a subtree can interact with a \(k\)-separator. One source states that for general \(k>2\), one would need to control all \(\binom{k}{i}\) ways that a subtree can meet a \(k\)-separator [1707.01165]. This suggests that the conjecture becomes increasingly sensitive to separator geometry as \(k\) grows. Within the current body of results, three tendencies are clear. First, exact proofs at the conjectured threshold are available for low \(k\) 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 [2101.11777], [2511.12499]. 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. Source: https://www.emergentmind.com/topics/mader-conjecture