We study generalizations of the Steiner Tree problem motivated by the design of power networks. While Steiner Tree asks for a single minimum-cost tree connecting given terminal vertices, a power network typically consists of multiple trees, each connecting a subset of the terminals, to avoid electrical overloads. The cost of installing depends on both the cable lengths and the cost of digging underground trenches for putting the cables where the digging costs can be shared. These leads to variants of Steiner Tree where the goal is to compute a minimum-cost set of Steiner trees with a common root, that together connect all terminals while balancing the power demand of the terminals in each tree. Two important variants arise depending on whether the network is intended for low-voltage or high-voltage power. In the low-voltage case, power loss imposes a bound on the maximum depth of each tree, while no such restriction applies in the high-voltage case. We study the parameterized complexity of several power network design problems, parameterized by the number of terminals. While Steiner Tree is fixed-parameter tractable under this parameterization, most of our variants are W[1]-hard. For low-voltage networks, we present an XP-algorithm for planar inputs based on structural bounds on the treewidth of solution subgraphs. We also give a reduction from Grid Tiling showing tightness under ETH. The XP-algorithm extends to the high-voltage setting and general graphs, albeit at a cost in the running time. For high-voltage networks, we show the problem remains W[1]-hard even on planar graphs. Finally, we explore a variant of the cost model for sharing digging costs in which both problems become fixed-parameter tractable.
Given a graph class 𝒢, the limiting density of 𝒢 is defined as δ(𝒢)=lim_n→∞𝖾𝗑(𝒢,n)/n where 𝖾𝗑(𝒢,n) is the maximum number of edges of a graph in 𝒢 on n vertices. The limiting density δ(𝒢) is known to be a rational number when 𝒢 is a minor-closed graph class. For every δ∈[0,3/2), we prove that the set of ⊆-minimal minor-closed graph classes with densities >δ is finite and we identify it completely. A consequence of our results is an algorithm that, given a finite set of graphs 𝒵, of total size n, either outputs the value of δ(𝖾𝗑𝖼𝗅(𝒵)) or reports that δ(𝖾𝗑𝖼𝗅(𝒵))≥3/2, where 𝖾𝗑𝖼𝗅(𝒵) is the class of graphs excluding the graphs in 𝒵 as minors. The algorithm runs in 2^𝗉𝗈𝗅𝗒(n) time.
We present the first rigorous analysis of decentralized geometric routing in Geometric Inhomogeneous Random Graphs (GIRGs), a weight-agnostic variant of the greedy routing protocol. While greedy routing in GIRGs is known to explain the algorithmic small-world phenomenon by finding ultra-short paths of length Θ(loglog n), it assumes additional knowledge of vertex weights beyond geometry, an assumption that is often restrictive or unavailable. We investigate whether the underlying geometry alone is sufficient for efficient navigation. We prove that for power-law weight exponent τ∈ (2,3) and geometric decay parameter α> τ- 1, geometric routing succeeds with constant probability and finds ultra-short paths of length Θ(loglog n), matching the optimal asymptotic guarantees for greedy routing. Our analysis further reveals that, upon success, both protocols follow a similar two-phase trajectory, consisting of a rapid ascent to the heavy vertices, followed by efficient navigation to the target. These results demonstrate that, in the appropriate regime, the network's geometry alone implicitly guides the path to the target through its high-weight core.
In Correlation Clustering, the input is a graph G=(V,E) with weight function ω: V 2→ Z and the task is to partition the vertex set into clusters such that the total weight of edges between clusters and missing edges inside clusters is minimized. Due to close connections between Correlation Clustering and Edge Multicut, deciding whether there is a partition with total cost at most k is FPT with respect to k but a polynomial kernel is presumably impossible. We study the influence of the structure of the fuzzy edge graph, that is, the graph induced by the weight-0 edges, on the problem complexity. We show in particular that Correlation Clustering admits a polynomial problem kernel when parameterized by k+d, where d is the degeneracy of the fuzzy edge graph, and when parameterized by k+c, where c is the closure of the fuzzy edge graph. We complement these positive results by showing hardness for several settings where the graph induced by the edges and nonedges has very restricted structure.
The conflict-free closed neighborhood (CFCN$^*$) chromatic number of a graph $G = (V,E)$ is the smallest positive integer $k$ for which there exists a coloring of a subset of vertices using $k$ colors such that, for every vertex in $V$, there exists a color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) chromatic number is defined analogously. In this paper, we study `list variants' of the above-mentioned coloring parameters. The conflict-free closed neighborhood (CFCN$^*$) choice number of a graph $G = (V,E)$ is the smallest positive integer $k$ such that for every assignment of lists of size $k$ to its vertices, there exists a coloring of a subset of vertices, say $V'$, in which (i) every vertex in $V'$ receives a color from its list, and (ii) for every vertex in $V$ there exists some color that appears exactly once in its closed neighborhood. The conflict-free open neighborhood (CFON$^*$) choice number is defined analogously. Dębski and Przybyło [Journal of Graph Theory, 2022] showed that for any graph $G$ with maximum degree $Δ$, the CFCN$^*$ chromatic number of its line graph is $O(\ln Δ)$. This result was later extended to claw-free graphs by Bhyravarapu et al. [Journal of Graph Theory, 2025], who proved that every $K_{1,k}$-free graph $G$ admits a CFCN$^*$ coloring using $O(k\ln Δ)$ colors. In this paper, we generalize this result to the list setting and show that every $K_{1,k}$-free graph $G$ has a CFCN$^*$ choice number of $O(k\ln Δ)$. Further, we answer some questions concerning the hardness of computing CFCN$^*$/CFON$^*$ choice numbers posed by Gupta and Mathew [SOFSEM, 2026]; in particular, we show that it is NP-hard to determine whether the CFCN$^*$/CFON$^*$ choice number a graph is equal to $k$, for $k=1,2$.
We consider upward-planar layered drawings of directed graphs, i.e., crossing-free drawings in which each edge is drawn as a y-monotone curve going upward from its tail to its head, and the y-coordinates of the vertices are integers. The span of an edge in such a drawing is the absolute difference between the y-coordinates of its endpoints, and the span of the drawing is the maximum span of any edge. The span of an upward-planar graph is the minimum span over all its upward-planar drawings. We study the problem of determining the span of upward-planar graphs and provide both combinatorial and algorithmic results. On the combinatorial side, we present upper and lower bounds for the span of directed trees. On the algorithmic side, we show that the problem of determining the span of an upward-planar graph is NP-complete already for directed trees and for biconnected single-source graphs. Moreover, we give efficient algorithms for several graph families with a bounded number of sources, including st-planar graphs and graphs where the planar or upward-planar embedding is prescribed. Furthermore, we show that the problem is fixed-parameter tractable with respect to the vertex cover number and the treedepth plus the span.
In this work we contribute to the study of the fine-grained complexity of problems parameterized by multi-clique-width, which was initiated by Fürer [ITCS 2017] and pursued further by Chekan and Kratsch [MFCS 2023]. Multi-clique-width is a parameter defined analogously to clique-width but every vertex is allowed to hold multiple labels simultaneously. This parameter is upper-bounded by both clique-width and treewidth (plus a constant), hence it generalizes both of them without an exponential blow-up. Conversely, graphs of multi-clique-width k have clique-width at most 2^k, and there exist graphs with clique-width at least 2^Ω(k). Thus, while the two parameters are functionally equivalent, the fine-grained complexity of problems may differ relative to them. As our first and main result we show that under ETH the Max Cut problem cannot be solved in time n^2^o(k)· f(k) on graphs of multi-clique-width k for any computable function f. For clique-width k an n^𝒪(k) algorithm by Fomin et al. [SIAM J. Comput. 2014] is tight under ETH. This makes Max Cut the first known problem for which the tight running times differ for parameterization by clique-width and multi-clique-width and it contributes to the short list of known lower bounds of form n^2^o(k)· f(k). As our second contribution we show that Hamiltonian Cycle and Edge Dominating Set can be solved in time n^𝒪(k) on graphs of multi-clique-width k matching the tight running time for clique-width. These results answer three questions left open by Chekan and Kratsch [MFCS 2023].
Given a connected undirected graph G, a spanning tree is a subgraph T of G such that V(T) = V(G) and T is a tree. A collection of ℓ spanning trees T_1,…,T_ℓ is pairwise k-diverse if for every i ≠ j, |E(T_i) E(T_j)| ≥ k. Given a connected undirected graph G and integers p, q, k, ℓ, Leaf Internal-Constrained Diverse Spanning Trees asks whether there are ℓ distinct spanning trees T_1,…,T_ℓ of G that are pairwise k-diverse such that each tree has at least p leaves and at least q internal vertices. Similarly, Leaf Non-terminal-Constrained Diverse Spanning Trees takes a connected undirected graph G, V_NT⊆ V(G), and three integers p, k, ℓ, and asks if G has ℓ spanning trees that are pairwise k-diverse, and each has at least p leaves and conains the vertices of V_NT as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for Leaf Internal-Constrained Diverse Spanning Trees and Leaf Non-terminal-Constrained Diverse Spanning Trees, when parameterized by p + q + k + ℓ and p + |V_ NT| + k + ℓ, respectively.
In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in H-free graphs. For Tight b-Chromatic Number in H-free graphs, we develop a general technique to determine new graphs H, for which the problem is polynomial-time solvable, and we also determine new graphs H, for which the problem is still NP-complete. We show, for the first time, the existence of a graph H such that in H-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.
The number of perfect matchings of a k-pfaffian graph can be counted by computing a linear combination of the pfaffians of k matrices. The pfaffian number of a graph G is the smallest integer k such that G is k-pfaffian. We present the first known lower bounds for the pfaffian number of graphs. As an intermediate step, we prove an upper bound for the rank of two matrices related to their Khatri-Rao product, a result of independent relevance. One of the consequences of these results is the existence of graphs whose pfaffian numbers are arbitrarily large.
Determining if an input undirected graph is Hamiltonian, i.e., if it has a cycle that visits every vertex exactly once, is one of the most famous NP-complete problems. We consider the following generalization of Hamiltonian cycles: for a fixed set S of natural numbers, we want to visit each vertex of a graph G exactly once and ensure that any two consecutive vertices can be joined in k hops for some choice of k ∈ S. Formally, an S-Hamiltonian cycle is a permutation (v_0,…,v_n-1) of the vertices of G such that, for 0 ≤ i ≤ n-1, there exists a walk between v_i and v_i+1 n whose length is in S. (We do not impose any constraints on how many times vertices can be visited as intermediate vertices of walks.) Of course Hamiltonian cycles in the standard sense correspond to S={1}. We study the S-Hamiltonian cycle problem of deciding whether an input graph G has an S-Hamiltonian cycle. Our goal is to determine the complexity of this problem depending on the fixed set S. It is already known that the problem remains NP-complete for S={1,2}, whereas it is trivial for S={1,2,3} because any connected graph contains a {1,2,3}-Hamiltonian cycle. Our work classifies the complexity of this problem for most kinds of sets S, with the key new results being the following: we have NP-completeness for S = {2} and for S = {2, 4}, but tractability for S = {1, 2, 4}, for S = {2, 4, 6}, for any superset of these two tractable cases, and for S the infinite set of all odd integers. The remaining open cases are the non-singleton finite sets of odd integers, in particular S = {1, 3}. Beyond cycles, we also discuss the complexity of finding S-Hamiltonian paths, and show that our problems are all tractable on graphs of bounded cliquewidth.
We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph optimally, Peroché [Discret. Appl. Math., 1984] proved that it is NP-hard already on graphs of maximum degree four, even when we only ask if two paths suffice. We show that the problem is solvable in polynomial time on subcubic graphs and then we present an efficient algorithm for “almost-subcubic” graphs. Precisely, we prove that the problem is fixed-parameter tractable when parameterized by the edge-deletion distance to a subcubic graph. To this end, we reduce the task to model checking in first-order logic extended by disjoint-paths predicates (𝖥𝖮+𝖣𝖯) and then we employ the recent tractability result by Schirrmacher, Siebertz, Stamoulis, Thilikos, and Vigny [LICS 2024].
For a directed graph G, and a linear order ≪ on the vertices of G, we define backedge graph G^≪ to be the undirected graph on the same vertex set with edge {u,w} in G^≪ if and only if (u,w) is an arc in G and w ≪ u. The directed clique number of a directed graph G is defined as the minimum size of the maximum clique in the backedge graph G^≪ taken over all linear orders ≪ on the vertices of G. A natural computational problem is to decide for a given directed graph G and a positive integer t, if the directed clique number of G is at most t. This problem has polynomial algorithm for t=1 and is known to be -complete for every fixed t≥3, even for tournaments. In this note we prove that this problem is Σ^𝖯_2-complete when t is given on the input.
Semialgebraic graphs are graphs whose vertices are points in ℝ^d, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs. That is, we show that for any family of semialgebraic graphs, given a graph on n vertices in this family, we can assign a label consisting of O(n^1-2/(d+1) + ε) bits to each vertex (where ε > 0 can be made arbitrarily small and the constant of proportionality depends on ε and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of O(n^1/3 + ε) bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of O(n^1-1/dlog n) for d-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-d polynomial. We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size O(log n). We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size O(log^3 n).
Hons, Klimošová, Kucheriya, Mikšaník, Tkadlec, and Tyomkyn proved that, for every integer ℓ≥ 1, every directed graph with minimum out-degree at least 3.23 ·ℓ contains a (2,ℓ)-spider (a 1-subdivision of the in-star with ℓ leaves) as a subgraph. They also conjectured that the bound on the minimum out-degree can be further improved to 2 ℓ. In this note, we confirm their conjecture by showing that every directed graph with minimum out-degree at least 2ℓ contains a (2, ℓ)-spider as a subgraph. This result is best possible, as the complete directed graph with 2ℓ vertices does not contain a (2,ℓ)-spider.
We introduce the tree-decomposition-based parameter totally Δ-modular treewidth (TDM-treewidth) for matrices with two nonzero entries per row. We show how to solve integer programs whose matrices have bounded TDM-treewidth when variables are bounded. This extends previous graph-based decomposition parameters for matrices with at most two nonzero entries per row to include matrices with entries outside of {-1,0,1}. We also give an analogue of the Grid Theorem of Robertson and Seymour for matrices of bounded TDM-treewidth in the language of rooted signed graphs.
