This paper proves that the former two problems are fixed-parameter tractable when parameterized by the treedepth of the input graph, and shows that Set of Degrees MST remains W[1]-hard parameterized by treedepth, even when combined with the feedback vertex number.
Abstract
We study the computation of minimum spanning trees subject to local degree constraints. Recent work (ICALP 2026) established that three natural formalizations of this problem share the exact same parameterized complexity under standard structural graph parameters, including treewidth, pathwidth and clique-width. This applies to the cases where every vertex has a single target degree (Specified Degree MST), or a degree upper bound (Bounded Degree MST), or is equipped with a set of admissible degrees (Set of Degrees MST). In this paper, we investigate these problems under more restrictive parameterizations and reveal that their complexity landscapes fundamentally diverge on bounded-treedepth graphs. Specifically, we prove that the former two problems are fixed-parameter tractable when parameterized by the treedepth of the input graph. In sharp contrast, we show that Set of Degrees MST remains W[1]-hard parameterized by treedepth, even when combined with the feedback vertex number (i.e., deletion distance to treewidth $1$). Finally, we show that this divergence seems to be specific to treedepth: we exclude an analogous W[1]-hardness result for Set of Degrees MST w.r.t. the vertex cover number and also rule out fixed-parameter algorithms for the former two problems w.r.t. deletion distance to constant pathwidth.
This paper asks what structural properties of the graph itself make metric repair tractable, and gives pseudo-polynomial time algorithms for series-parallel graphs, and by generalization, graphs of bounded treewidth and a new algorithm for the length-bounded multicut problem.
Asaf Etgar, A. Gilbert, Jamie Tucker-Foltz· arXiv.org· 0 citations
A graph is outer k-planar if it admits a straight-line drawing in which all vertices lie on a circle and every edge is crossed by at most k other edges. We study the parameterized complexity of a broad collection of graph problems on outer k-planar graphs, with k as the parameter. Many graph problems are known to be XALP-hard when parameterized by treewidth or outerplanarity, and XNLP-hard when parameterized by pathwidth. We show that only a few such problems, including Binary CSP and Scattered Set, remain intractable on outer k-planar graphs, whereas a large class of the others become fixed-parameter tractable in this setting, assuming that an outer k-planar drawing of the input graph is given. These include List Coloring, Capacitated Dominating Set, Capacitated Vertex Cover, Target Outdegree Orientation, and Target Set Selection, among others. In addition to the algorithmic and complexity results, we establish several structural results. We show that outer k-planar graphs have mim-width at most k+2, that graphs of cut-width at most k are outer 2k-planar, and that graphs of feedback edge set number at most k are outer 6k-planar. We also show that many graph parameters are incomparable with outer k-planarity, thereby clarifying its position within the graph parameter hierarchy.
Xiaobin Ren, H. Bodlaender· arXiv.org· 0 citations
We study the parameterized complexity of Induced Subgraph Isomorphism (ISI) and Maximum Common Induced Subgraph (MCIS) with respect to the cluster vertex deletion number $k$. For ISI, we give a randomized $O^*(k^{O(k)})$-time algorithm, showing that ISI is fixed-parameter tractable under this parameter and resolving an open question of Hanaka et al. [WALCOM 2026]. Our algorithm is optimal under the Exponential Time Hypothesis (ETH), and is based on a reduction to Exact Multicolored Matching solvable via algebraic techniques. For MCIS, we present a randomized $O^*(2^{O(k^2)})$-time algorithm via a reduction to a weighted variant of Exact Multicolored Matching, and we prove a matching ETH-based lower bound by showing that a $k$-by-$k$ binary matrix feasibility problem with list-constrained rows and columns admits no $O^*(2^{o(k^2)})$-time algorithm, which may be of independent interest. These results reveal that, in this setting, MCIS is strictly harder than ISI. Finally, for the three-graph variant 3-MCIS, we show that it becomes NP-hard already when each input graph has cluster vertex deletion number 2.
Tomohiro Koana, Soh Kumabe, Y. Otachi· 0 citations
It is proved that the classical cut property always produces a minimum spanning tree of a connected graph, and may be useful for large weighted networks such as communication networks, wiring connections, and transportation networks.
H. Bhapkar, Rezwan Ul Shaban, S. Mir et al.· Journal of the Nigerian Soci...· 0 citations
This paper generalises the existing PTAS for complete graphs from a fixed to an arbitrary number of intervals and disprove an existing conjecture, which states that every instance of BIA admits a solution satisfying at least three quarters of all edges.
∗. PeterBlohm, ∗. FlorianChen, A. Gionis et al.· 0 citations
