Disjoint Connectivity Keeping Trees

• Disjoint connectivity keeping trees are subtrees with strict disjointness (edge-, vertex-, or internally disjoint) that maintain connectivity and bolster network reliability.
• They employ advanced graph-theoretical methods, including spectral analysis and combinatorial optimization, to determine connectivity thresholds and packing limits.
• Applications include network fault tolerance, distributed algorithms, and reliable broadcasting, with significant focus on algorithmic complexity and NP-completeness challenges.

A disjoint connectivity keeping tree is a subtree structure with a highly constrained disjointness property, central to the paper of network reliability, generalized connectivity, and resilience. In its most common forms, it refers to a set of connectivity-maintaining (e.g., spanning, Steiner, or pendant-Steiner) trees that are either edge-disjoint, vertex-disjoint, internally disjoint, or satisfy weaker/stronger joint intersection constraints, with variants in both undirected and directed graphs. The packing, construction, and extremal limits of such trees—especially their generalized $k$-connectivity and algorithmic complexity—constitute a vibrant area at the intersection of graph theory, combinatorial optimization, and distributed algorithms.

1. Foundational Notions and Disjointness Variants

A connectivity keeping tree in a graph $G=(V,E)$ connects a specified set of terminals $S\subseteq V$ (often all of $V$, yielding a spanning tree), such that removal of its vertices/edges from $G$ preserves or achieves some connectivity property. The strictest form is internally disjoint $S$-Steiner trees: for $|S|\ge2$, $T_1,\dots,T_\ell$ are pairwise edge-disjoint and $V(T_i)\cap V(T_j)=S$ for all $i\neq j$ (Li et al., 2010, Li et al., 2012). Edge-disjointness, vertex-disjointness, and pendant (all terminals have degree 1) variants are also prominent (Mao, 2015, Yu et al., 1 May 2025).

In directed graphs, the analogous object is an internally disjoint out-tree: each tree is rooted at a terminal $r\in S$, covers $S$, and trees only overlap at $S$ (Sun et al., 2020, Sun, 2020).

A generalization is the $(i,j)$-disjoint spanning tree framework, quantifying the maximum number of common inner vertices ($i$) and edges ($j$) permitted between trees, interpolating between edge-disjoint, vertex-disjoint, and the stringent completely independent spanning trees where both $i=j=0$ (Darties et al., 2017).

2. Generalized Connectivity and Packing Numbers

The central invariant is the generalized $k$-connectivity: $$\kappa_k(G) = \min\{\kappa_G(S) : S\subseteq V(G),\ |S|=k\}$$ where $\kappa_G(S)$ is the maximum number of internally disjoint $S$-Steiner trees (Li et al., 2010, Yang et al., 2023). Analogously, the generalized $k$-edge-connectivity $\lambda_k(G)$ is defined via edge-disjointness (Yang et al., 2023). For directed graphs, these parameters extend to $\kappa_k(D)$ and $\lambda_k(D)$ via out-trees (Sun et al., 2020, Sun, 2020):

$$\kappa_k(D) = \min\{\kappa_{S,r}(D): S\subseteq V, |S|=k, r\in S\}$$

$$\lambda_k(D) = \min\{\lambda_{S,r}(D): S\subseteq V, |S|=k, r\in S\}$$

These are the primary group-based resilience measures in the presence of vertex or edge failures.

The classical case $k=2$ recovers usual (vertex/edge) connectivity. For $k\ge3$, computing $\kappa_k(G)$ or $\kappa_{S, r}(D)$ becomes substantially more challenging, often NP-complete (Li et al., 2010, Sun et al., 2020).

3. Extremal, Spectral, and Algorithmic Results

Min-Max Packing and Tightness

The Tutte-Nash-Williams theorem gives the exact packing number $\tau(G)$ of edge-disjoint spanning trees in undirected graphs: $$\tau(G) = \min_{P} \left\lfloor \frac{|E(P)|}{|P|-1} \right\rfloor$$ with $P$ ranging over all partitions of $V$ (Bailey et al., 2010, Chandrasekaran et al., 25 Mar 2025). Equality $\tau(G) = \lambda(G)$ (where $\lambda$ is minimum edge-cut size) precisely characterizes maximum packable cases; such graphs decompose into $\lambda$-irreducible components along tight edge-cuts (Bailey et al., 2010).

Spectral Methods

Spectral bounds connect adjacency and signless Laplacian eigenvalues to edge-connectivity and tree packing: for $G$ in a suitable family $\mathcal{G}$, if the third-largest adjacency eigenvalue $\lambda_3(G)$ and signless Laplacian eigenvalue $q_3(G)$ satisfy explicit inequalities parameterized by minimum/maximum degrees $\delta, \Delta$ and the target packing $k$, then packing and connectivity thresholds hold (see Table):

Condition	Spectral Bound (Adjacency)	Spectral Bound (Signless Laplacian)
$\kappa'(G)\ge k$	$$\lambda_3(G) < 2\delta - \Delta - \frac{4(k-1)}{\delta+1}$$	$$q_3(G) < 4\delta - 2\Delta - \frac{4(k-1)}{\delta+1}$$
$\tau(G)\ge k$	$$\delta\ge 2k,\ \lambda_3(G)<2\delta-\Delta-\frac{2(3k-1)}{\delta+1}$$	$$\delta\ge 2k,\ q_3(G)<4\delta-2\Delta-\frac{2(3k-1)}{\delta+1}$$

This framework uses quotient matrices and eigenvalue interlacing to force the minimum cut to be large enough to ensure $k$ edge-disjoint trees (Duan et al., 2017).

Complexity and Algorithms

• For fixed $k_1, k_2$, determining whether $G$ contains $k_2$ internally disjoint $S$-trees for a fixed $S$, or $\kappa_k(G) \ge k_2$, is polynomial-time solvable, leveraging the bounded number of tree isomorphism types and Robertson–Seymour linkage algorithms (Li et al., 2010).
• NP-completeness arises as soon as either $k_1$ (the terminal set size) or $k_2$ (the number of trees) is variable, even for moderate parameter values (Li et al., 2010, Sun et al., 2020).
• Online and distributed models: various approximation algorithms with $O(\log^2 n)$ competitive ratios for online packing of disjoint spanning trees (viewed as packing polymatroid bases), using randomized coloring and quotient techniques (Chandrasekaran et al., 25 Mar 2025). In distributed models, fast $\widetilde{O}(D+\sqrt{n \lambda})$-round algorithms achieve close to optimal fractional tree packings and support low-congestion routing (Censor-Hillel et al., 2013).

4. Specialized Constructions and Extremal Results

Sierpiński and Product Graphs

In recursive Sierpiński graphs $S(n, \ell)$, explicit formulas for generalized connectivity are available: $$\kappa_k(S(n,\ell)) = \lambda_k(S(n,\ell)) = \begin{cases} \ell - \lfloor k/2 \rfloor, & 3 \leq k \leq \ell \ \lfloor \ell/2 \rfloor, & k > \ell \end{cases}$$ Recursive constructions exploit atom partitions and Hamiltonian path decompositions to realize the maximal packing (Yang et al., 2023).

For Cartesian products $G \Box H$, exact lower bounds on local pendant tree-connectivity are obtained: for $k=3$, $$\tau_3(G \Box H) \geq \min\{3 \lfloor \tau_3(G)/2 \rfloor, 3 \lfloor \tau_3(H)/2 \rfloor \}$$ (Mao, 2015).

Networks with Triangle- or Cycle-Free Constraints

In triangle-free graphs, high minimum degree (explicitly, $\delta(G) \ge 2k + 3m - 4$ for a $k$-connected graph and $m$-vertex tree $T$) ensures existence of a $T'$ whose removal preserves $k$-connectivity (Chu et al., 10 Nov 2025). Girth and bipartite assumptions further sharpen the degree thresholds.

Extremal Edge Counts

For $k=n$ (spanning trees), maximal edge counts with at most $\ell$ edge-disjoint trees are characterized by

$$f(n;\overline\kappa_n \leq \ell) = \binom{n-1}{2} + \ell$$

and analogs for $k=n-1$ and general $k$, with extremal graphs constructed by connecting all but one vertex in a large clique and attaching smaller structures (Li et al., 2013, Li et al., 2012).

5. Directed Graphs: Out-Trees and Pendant Connectivity

Directed versions require careful adaptation:

• Directed generalized tree connectivity $\kappa_{k}(D)$ considers the minimum number of internally disjoint out-trees rooted at a specified terminal subset $S$, sharing vertices only in $S$ (Sun et al., 2020, Sun, 2020).
• Pendant-tree connectivity $\tau_{k}(D)$ measures the minimum number of internally disjoint directed out-trees with all terminals as leaves; undirected and directed cases differ in both bounds and computational complexity (Yu et al., 1 May 2025).

Sharp upper bounds $\tau_k(D) \le n-k$, and tight min-cut based inequalities are available, fully attainable in complete symmetric digraphs.

NP-completeness of packing pendant-trees or more general out-trees is the rule on Eulerian digraphs, but for symmetric digraphs and fixed parameters, polynomial-time results hold (Yu et al., 1 May 2025).

6. Applications and Interpretations

Disjoint connectivity keeping trees are fundamental in:

• Network reliability: maximizing the number of edge-disjoint or vertex-disjoint spanning/Steiner/multicast trees corresponds directly to multi-route communication resilience, fault-tolerant broadcasting, and capacity provisioning (Censor-Hillel et al., 2013, Bailey et al., 2010, Hoyer et al., 2017).
• Distributed and online systems: distributed decomposition and tree-packing underpins optimal message routing, parallel communication, and network repair strategies (Censor-Hillel et al., 2013, Chandrasekaran et al., 25 Mar 2025).

Network models also benefit from fine-grained $(i,j)$-disjointness, interpolating between strict and loose redundancy requirements (Darties et al., 2017).

7. Open Problems and Future Directions

Open questions include:

• Precise determination of extremal functions $f(n; \overline\kappa_k \le \ell)$ for $k\ge4$ or arbitrary $\ell$, and characterization of extremal graph families (Li et al., 2012, Li et al., 2013).
• Reducing degree thresholds for connectivity-keeping trees in triangle-free or $P_\ell$-free graphs, and resolving conjectured lower bounds (Chu et al., 10 Nov 2025, Hasunuma, 16 Nov 2025).
• Achieving $O(1)$-competitive ratios for online tree packing, or extending structural decompositions to broader matroid classes (Chandrasekaran et al., 25 Mar 2025).
• Full complexity dichotomy for directed pendant/Steiner packing—especially for variable parameter regimes (Yu et al., 1 May 2025, Sun et al., 2020).

These research avenues indicate the deep interplay between combinatorial structure, algorithmics, and extremal graph theory in the paper of disjoint connectivity keeping trees.