We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph $G$ and a constraint function $D$, we ask for a (minimum-cost) spanning tree $T$ such that for each vertex $v$, $T$ achieves a degree specified by $D(v)$. Specifically, we consider three kinds of constraint functions ordered by their generality -- $D$ may either assign each vertex to a list of admissible degrees, an upper bound on the degrees, or a specific degree. Using a combination of novel techniques and state-of-the-art machinery, we obtain an almost-complete overview of the fine-grained complexity of these problems taking into account the most classical graph parameters of the input graph $G$. In particular, we present SETH-tight upper and lower bounds for these problems when parameterized by the pathwidth and cutwidth, an ETH-tight algorithm parameterized by the cliquewidth, and a nearly SETH-tight algorithm parameterized by treewidth. In order to obtain our upper bound for clique-width, we develop a novel technique of double representation through ``requirement shifting''. Using this technique, we also obtain an ETH-tight single-exponential XP algorithm for the Exact Leaf Spanning Tree problem parameterized by clique-width, which settles the final remaining open case for clique-width from the classical Cut and Count of Cygan et al. [FOCS 2011, TALG 2022]. This shows the versatility of our technique and its potential applicability to other problems as well. Additionally, in order to establish our lower and upper bounds we introduce a number of tools which may be of independent interest, including lazy coloring and ``asymptotic'' SETH-based reductions for structural parameters.

Given a planar graph, a subset of its vertices called terminals, and k ∈ ℕ, the Face Cover Number problem asks whether the terminals lie on the boundaries of at most k faces of some embedding of the input graph. When a plane graph is given in the input, the problem is known to have a polynomial kernel [Valentin Garnero et al., 2017]. In this paper, we present the first polynomial kernel for Face Cover Number when the input is a planar graph (without a fixed embedding). Our approach overcomes the challenge of not having a predefined set of face boundaries by building a kernel bottom-up on an SPR-tree while preserving the essential properties of the face cover along the way.

The class XNLP consists of (parameterized) problems that can be solved non-deterministically in f ( k ) n O ( 1 ) time and g ( k ) log n space, where n is the size of the input instance and k the parameter. The class XALP consists of problems that can be solved in the above time and space with access to an additional stack. These two classes are a “natural home” for many standard graph problems and their generalizations. In this paper, we show the hardness of several problems on planar graphs, parameterized by outerplanarity, treewidth and pathwidth, thus strengthening several existing results. In particular, we show XALP-completeness of the following problems parameterized by outerplanarity: All-or-Nothing Flow , Target Outdegree Orientation , Capacitated (Red–Blue) Dominating Set , Target Set Selection etc. We also show the XNLP-completeness of Scattered Set parameterized by pathwidth and XALP-completeness parameterized by treewidth and outerplanarity.

In this paper, we study the problem of pathfinding on traversal-dependent graphs, i.e., graphs whose edges change depending on the previously visited vertices. In particular, we study self-deleting graphs, introduced by Carmesin et al. [Sarah Carmesin et al., 2023], which consist of a graph G = (V, E) and a function f: V → 2^E, where f(v) is the set of edges that will be deleted after visiting the vertex v. In the (Shortest) Self-Deleting s-t-path problem we are given a self-deleting graph and its vertices s and t, and we are asked to find a (shortest) path from s to t, such that it does not traverse an edge in f(v) after visiting v for any vertex v. We prove that Self-Deleting s-t-path is NP-hard even if the given graph is outerplanar, bipartite, has maximum degree 3, bandwidth 2 and |f(v)| ≤ 1 for each vertex v. We show that Shortest Self-Deleting s-t-path is W[1]-complete parameterized by the length of the sought path and that Self-Deleting s-t-path is W[1]-complete parameterized by the vertex cover number, feedback vertex set number and treedepth. We also show that the problem becomes FPT when we parameterize by the maximum size of f(v) and several structural parameters. Lastly, we show that the problem does not admit a polynomial kernel even for parameterization by the vertex cover number and the maximum size of f(v) combined already on 2-outerplanar graphs.

We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small \emph{edge} separators. Let $p,q \in \mathbb{N}$ such that $p$ is a prime and $q \geq 3$. - If $p$ divides $q-1$, there is a $(q-1)^{\text{ctw}}n^{O(1)}$ time algorithm for counting list $q$-colorings modulo $p$ of $n$-vertex graphs of cutwidth $\text{ctw}$ and for all $\varepsilon>0$ there is no algorithm running in time $(q-1-\varepsilon)^{\text{ctw}} n^{O(1)}$, assuming the Strong Exponential Time Hypothesis (SETH). - If $p$ does not divide $q-1$, there is a (folklore) $q^{\text{ctw}}n^{O(1)}$ time algorithm for counting list $q$-colorings modulo $p$ of $n$-vertex graphs of cutwidth $\text{ctw}$ and for all $\varepsilon>0$ there is no algorithm running in time $(q-\varepsilon)^{\text{ctw}} n^{O(1)}$, assuming SETH. The lower bounds are in stark contrast with the existing $2^{\text{ctw}}n^{O(1)}$ time algorithm to compute the chromatic number of a graph by Jansen and Nederlof~[Theor. Comput. Sci.'18]. Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain `compatibility matrix' in a non-trivial way. We extend our lower bounds to counting connected spanning edge sets modulo $p$ and give an algorithm with matching running time for both treewidth and cutwidth.