construct independent spanning trees

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 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 work addresses the path cover problem on undirected graphs: determining whether the graph admits a vertex-disjoint path cover of fewer than α(G) − k paths, where α(G) is the independence number and k is a parameter. We establish the first fixed-parameter tractable (FPT) framework parameterized by the *independence-number deviation* k—leveraging graph density rather than conventional sparsity measures. Our method introduces novel techniques: an independent-set-guided search tree with aggressive pruning, combined with dynamic programming and subgraph enumeration, yielding a 2^{k^{O(k⁴)}}·n^{O(1)}-time algorithm. We extend the Gallai–Milgram theorem to an FPT setting for the first time; provide a polynomial-time algorithm for Hamiltonian path detection in graphs with α(G) ≤ 3; and unify the FPT complexity of five NP-hard problems—including Hamiltonian cycle, path cover, and longest path—under the independence number parameter, thereby overcoming limitations inherent to treewidth and other sparsity-based parameters.

This paper resolves the complexity classification of the Maximum Weight Independent Set (MWIS) problem on $H$-free graphs, where every connected component of $H$ is a path or a subdivided claw. It establishes, for the first time, that MWIS admits a quasipolynomial-time algorithm—running in time $|V|^{O(log |V|)}$—on $H$-free graphs for *every* such $H$, thereby completing the dichotomy for $F$-free graphs: all previously open cases are now shown to be tractable. The key technical innovation is a strengthened structural lemma enabling a weight-decreasing extended strip decomposition *without* preprocessing vertex deletions. This is combined with advanced tools from structural graph theory, quasipolynomial branching, balanced separators, and detection of induced $S_{t,t,t}$ subgraphs. The result significantly extends the boundary of efficiently solvable MWIS instances and provides crucial evidence supporting the conjecture that all non-NP-hard $H$-free graph classes lie in P.

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 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 investigates the applicability limits of irrelevant-vertex techniques for the disjoint paths problem with crossing constraints. Focusing on the distribution of terminal vertices in labeled graphs, the authors introduce a new structural parameter, depth₂, and prove that it precisely characterizes the effectiveness of irrelevant-vertex rules: when depth₂ is bounded, a fixed-parameter algorithm exists with running time $2^{2^{\text{poly}(k+d)}} \cdot n^2$; however, if depth₂ is unbounded, no such rule exists even on planar graphs. By integrating graph minor theory and structural graph theory, the work establishes a local structure theorem for graph classes with bounded depth₂ and a key connectivity theorem for routed path variants, leading to novel irrelevant-vertex reduction rules and decomposition methods tailored to labeled graphs.

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 addresses the problem of determining whether two pairs of vertex sets in a graph admit disjoint separators, which is equivalent to the existence of a red-blue coloring such that no red path connects vertices within the red sets and no blue path connects vertices within the blue sets. The problem carries both theoretical significance and applications in game theory, notably in establishing draw-free conditions for generalized Hex boards. Through graph-theoretic modeling, complexity reductions, and structural decomposition, this work provides the first systematic characterization of its computational complexity: it proves NP-completeness for general graphs, planar graphs, and bounded-degree graphs. However, when the graph is planar and each set consists of a single vertex, the paper identifies a structural characterization and presents a polynomial-time decision algorithm.

This study investigates upper bounds on the tree-independence number of graph classes closed under induced subgraphs and their algorithmic implications. By leveraging graph decomposition theory, induced subgraph analysis, and extremal combinatorial methods, it is shown that any graph excluding a complete bipartite graph $K_{t,t}$ or a grid graph $W_{t\times t}$ as an induced subgraph has subpolynomial tree-independence number. This result yields the first trichotomy for such graph classes based on their tree-independence number—linear, square-root, or subpolynomial—and partially resolves a conjecture by Chudnovsky et al. regarding polylogarithmic bounds. The derived bound of $O(2^{O((\log n)^{1-\varepsilon})})$ leads to a universal algorithm running in time $2^{n^{o(1)}}$, applicable to a broad range of graph optimization problems.