construct completely independent spanning trees

This paper investigates the existence of Completely Independent Spanning Trees (CISTs)—i.e., two edge-disjoint and internally vertex-disjoint spanning trees—in split graphs. Methodologically, it establishes the first exact equivalence between CIST existence and two hypergraph coloring notions: polychromatic and bichromatic colorings—thereby uncovering a deep structural connection between graph topology and hypergraph coloring. Leveraging this correspondence, the authors formulate and validate an original conjecture on the bichromatic number of split graphs. Through combinatorial modeling, structural graph analysis, and NP-completeness reductions, they derive tight upper and lower bounds on the maximum number of CISTs and prove that deciding the existence of two CISTs in split graphs is NP-complete. The principal contribution is a rigorous equivalence framework linking CIST existence to polychromatic and bichromatic hypergraph colorings—yielding the first structural characterization and computational complexity classification for CISTs in split graphs.

This study investigates the existence of two completely independent spanning trees (CISTs) in $k$-outerplanar triangulations. Through path-disjointness analysis, structural induction, and constructive proofs, it establishes for the first time that every 3-connected 2-outerplanar triangulation admits two CISTs. For the 3-outerplanar case, the work provides a sufficient condition guaranteeing the existence of such trees. Moreover, it constructs the first known counterexample—a 4-outerplanar triangulation that does not contain two CISTs. These results precisely delineate the boundary for CIST existence in low-level outerplanar graphs, significantly advancing the structural understanding of this important graph class.

This paper investigates vertex-separating path/tree systems on graphs: families of paths or subtrees such that every pair of vertices is contained in exactly one member—achieving exact separation. We establish the first systematic characterization of the minimum size of such systems for three fundamental graph classes: Θ(log n) for trees, Θ(n) for n×n grid graphs, and Θ(n) for maximal outerplanar graphs. Using combinatorial graph theory, recursive decomposition, extremal constructions, and duality arguments, we derive tight upper and lower bounds and provide matching explicit constructions—achieving both theoretical optimality and constructibility. The core contribution is the introduction and resolution of the “exact vertex separation” model, a novel combinatorial framework that uncovers an intrinsic connection between graph structural complexity and separation efficiency.

This paper studies the *tree-independence number* τ(G)—the maximum independence number among the subgraphs induced by the bags of an optimal tree decomposition of G. This parameter enables efficient algorithms for NP-hard problems such as Maximum Weight Independent Set (MWIS). We present the first fixed-parameter approximation algorithm for τ(G): it computes an 8k-approximation in time $2^{O(k^2)} n^{O(k)}$. We prove that this runtime is conditionally tight under Gap-ETH, and show that exact computation of τ(G) is para-NP-hard for k ≥ 4. This yields the first complete parameterized complexity characterization of τ(G), resolving its approximability and hardness without assuming a precomputed tree decomposition. Moreover, our result implies a $2^{O(k^2)} n^{O(k)}$-time algorithm for MWIS on graphs with τ(G) ≤ k—bypassing the traditional requirement of being given a tree decomposition with bounded independence number.

This paper studies the NP-hard Steiner Tree Congestion (STC) optimization problem: constructing a spanning tree that minimizes the maximum number of times any original graph path traverses a single tree edge. For sparse graphs with $m = O(n log n)$ edges and bounded maximum degree $Delta$, we establish the first tight lower bound relating STC to the hereditary bisection width $hb(G)$: $mathrm{STC}(G) geq Omega(hb(G)/Delta)$. Leveraging this bound, we design an $O(Delta cdot log^{3/2} n)$-approximation algorithm. Our method integrates hereditary bisection width analysis, recursive graph partitioning, and degree-constrained lower-bound derivation. For graphs with $Delta leq mathrm{polylog}(n)$, the algorithm achieves an $O(log^{3/2} n)$ approximation ratio—breaking the classical $O(n)$ barrier—and constitutes the first STC approximation algorithm whose performance guarantee is sublinear in $Delta$. This yields significantly improved theoretical guarantees and constructive techniques for low-degree sparse graphs.

This study presents the first systematic investigation of the Completely Independent Steiner Tree (CIST) problem, which seeks to construct multiple Steiner trees for fault-tolerant communication such that the internal vertices and edges of the paths between terminals are mutually disjoint across trees. The authors unify the notions of completely independent spanning trees and internally disjoint Steiner trees, introduce a directed variant, and establish their equivalence. Through graph-theoretic analysis, structural characterizations, and complexity-theoretic arguments—complemented by dynamic programming and specialized algorithms for planar and bounded-treewidth graphs—the work proves the NP-hardness of the problem in general graphs, establishes a lower bound on graph connectivity sufficient for CIST existence, and designs polynomial-time algorithms for several restricted graph classes.

This study addresses the insufficient fault tolerance and communication reliability of dense Gaussian networks in routing and broadcasting scenarios. To overcome these limitations, the authors propose a novel construction method for two completely independent spanning trees (CISTs). By partitioning the node set and incorporating a rotation mechanism, the approach efficiently generates two structurally optimized CISTs, with the second tree exhibiting reduced depth. Compared to existing techniques, the proposed scheme significantly enhances both fault tolerance and communication efficiency, reducing the average maximum number of hops required for a message to propagate from the root to all nodes in the network by at least 33%.

This study resolves a long-standing open problem concerning the existence of two completely independent spanning trees in the four-dimensional folded cube $F_4$. By explicitly constructing a cubic polynomial with ten terms over the finite field $\mathbb{F}_2$, the authors define internal vertices via its level sets and employ a solver-free program to verify connectivity, thereby successfully building two completely independent spanning trees using 254 edges. This result completes the characterization that $F_n$ admits two completely independent spanning trees if and only if $n \geq 4$. Moreover, it establishes the optimality of the construction by proving that neither affine nor quadratic polynomials suffice within this algebraic framework, and that ten terms constitute the minimal number required for any cubic rule.

This study addresses the problem of efficiently enumerating all tuples of $k$ edge-disjoint spanning trees in a graph. Building upon Kaiser’s constructive proof of the Tutte–Nash-Williams theorem, the authors propose a recursive algorithm that integrates decision tree construction, depth-first search, and a variant of Kaiser’s technique, dynamically maintaining a forest-packing structure to decompose subproblems. This work presents the first polynomial-delay enumeration algorithm for tuples of edge-disjoint spanning trees, thereby transforming a classical structural result in graph theory into a practical and efficient enumeration method. The algorithm guarantees that all feasible solutions are output completely, with only polynomial delay between successive outputs.

This study investigates the computational complexity of the Steiner Tree problem on split-like graph classes, including bipartite, split, and bisplit graphs. By introducing a unified framework for split-like graphs and leveraging reductions from Exact-3-Cover alongside structural graph analysis, the work precisely delineates the boundary between polynomial-time solvability and NP-completeness. Key contributions include establishing that the problem is polynomial-time solvable in $K_{1,r}$-free bipartite graphs when $r \leq 3$ but becomes NP-complete for $r \geq 4$; demonstrating efficient solvability on bisplit graphs of diameter 2, while proving NP-completeness for diameters 3 and 4; and providing a complete complexity classification under constraints of star-convexity and chordality.