Algorithmic meta-theorems explain the tractability of large classes of computational problems by linking logical expressibility with structural graph properties. While extensions of first-order logic such as FO+dp admit efficient model checking on graph classes excluding a fixed topological minor, comparable results for richer fragments of CMSO were previously unknown. We further develop the framework of Sau, Stamoulis, and Thilikos [SODA 2025] for fragmenting CMSO via annotated graph parameters, which restrict set quantification to vertex sets satisfying bounded structural conditions. Following this approach, we identify a fragment of CMSO, namely the one defined by allowing quantification only over sets having what we call low monodimensionality, that generalizes several previously-known logics and we show that model checking for this fragment, enhanced with the disjoint-paths predicate, is fixed-parameter tractable on topological-minor-free graph classes. Such classes essentially delimit the tractability for this logic on subgraph-closed classes. As a consequence, our results lift several known algorithmic meta-theorems beyond first-order logic to the topological-minor-free setting.

We study solution discovery, where the goal is to obtain a feasible solution to a problem from an initial configuration by a bounded sequence of local moves. In many applications, however, the graph that defines which vertex sets are feasible is not the same as the graph that governs how tokens, agents, or resources may move. Existing models such as token sliding and token jumping typically do not distinguish the problem graph and the movement graph. Motivated by this mismatch, we introduce a directed weighted two-graph model that cleanly separates feasibility from movement. A problem graph specifies the desired combinatorial objects, while a movement graph specifies admissible relocations and their costs. This yields a flexible framework that captures asymmetry, heterogeneous movement constraints, and weighted transitions, while subsuming classical discovery models as special cases. We investigate this model through Path Discovery and Shortest Path Discovery, where the task is to realize a vertex set containing an s-t-path or a shortest s-t-path in the problem graph. These problems are particularly natural in applications, since directed and weighted shortest paths are among the most fundamental algorithmic primitives. At the same time, previous work has already shown that discovery can be computationally hard even when the underlying optimization problem is easy. Our results show that this phenomenon persists, and becomes especially rich, in the two-graph setting. We obtain a detailed complexity picture, identifying tractable cases as well as strong hardness results.

In the Dominated Cluster Deletion problem, we are given an undirected graph G and integers k and d and the question is to decide whether there exists a set of at most k vertices whose removal results in a graph in which each connected component has a dominating set of size at most d. In the Elimination Distance to Dominated Clusters problem, we are again given an undirected graph G and integers k and d and the question is to decide whether we can recursively delete vertices up to depth k such that each remaining connected component has a dominating set of size at most d. Bentert et al. [Bentert et al., MFCS 2024] recently provided an almost complete classification of the parameterized complexity of Dominated Cluster Deletion with respect to the parameters k, d, c, and Δ, where c and Δ are the degeneracy, and the maximum degree of the input graph, respectively. In particular, they provided a non-uniform algorithm with running time f(k,d)⋅ n^{𝒪(d)}. They left as an open problem whether the problem is fixed-parameter tractable with respect to the parameter k + d + c. We provide a uniform algorithm running in time f(k,d)⋅ n^{𝒪(d)} for both Dominated Cluster Deletion and Elimination Distance to Dominated Clusters. We furthermore show that both problems are FPT when parameterized by k+d+𝓁, where 𝓁 is the semi-ladder index of the input graph, a parameter that is upper bounded and may be much smaller than the degeneracy c, positively answering the open question of Bentert et al. We further complete the picture by providing an almost full classification for the parameterized complexity and kernelization complexity of Elimination Distance to Dominated Clusters. The one difficult base case that remains open is whether Treedepth (the case d = 0) is NP-hard on graphs of bounded maximum degree.

Disjoint-paths logic, denoted FO+ DP, extends first-order logic (FO) with atomic predicates dpk[(x1, y1), …, (xk, yk)], expressing the existence of internally vertex-disjoint paths between xi and yi, for 1 ≤ i ≤ k. We prove that for every graph class excluding some fixed graph as a topological minor, the model checking problem for FO+ DP is fixed-parameter tractable. This extends the model checking algorithm of Golovach et al. [SODA 2023] for FO+ DP for minor-closed graph classes. It also essentially settles the question of tractable model checking for this logic on subgraph-closed classes, since the problem is hard on subgraph-closed classes not excluding a topological minor (assuming a further mild condition on efficiency of encoding).

First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. However, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph problems that are commonly studied in parameterized algorithmics. By adding the atomic predicates connk(x,y,z_1,..., zk) that hold true in a graph if there exists a path between (the valuations of)xandyafter (the valuations of)z1,..., zkhave been deleted, we obtainseparator logicFO + conn. We show that separator logic can express many interesting problems, such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO + connkthe fragment of separator logic that is restricted to connectivity predicates with at mostk + 2variables (that is, at mostkdeletions), we show that FO + connk + 1is strictly more expressive than FO + connkfor allk ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the strongerdisjoint-paths logicFO + DP by adding the atomic predicates disjoint-pathsk[(x1, y1),..., (xk, yk) that evaluate to true if there are internally vertex-disjoint paths between (the valuations of)xiandyifor all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again, we show that the fragments FO + DPkthat use predicates for at mostkdisjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.

First-order logic (FO) can express many algorithmic problems on graphs, such as the independent set and dominating set problem parameterized by solution size. On the other hand, FO cannot express the very simple algorithmic question whether two vertices are connected. We enrich FO with connectivity predicates that are tailored to express algorithmic graph properties that are commonly studied in parameterized algorithmics. By adding the atomic predicates conn_k(x,y,z_1,…,z_k) that hold true in a graph if there exists a path between (the valuations of) x and y after (the valuations of) z_1,…,z_k have been deleted, we obtain separator logic FO+conn. We show that separator logic can express many interesting problems such as the feedback vertex set problem and elimination distance problems to first-order definable classes. Denote by FO+conn_k the fragment of separator logic that is restricted to connectivity predicates with at most k+2 variables (that is, at most k deletions). We show that FO+conn_{k+1} is strictly more expressive than FO+conn_k for all k ≥ 0. We then study the limitations of separator logic and prove that it cannot express planarity, and, in particular, not the disjoint paths problem. We obtain the stronger disjoint-paths logic FO+DP by adding the atomic predicates disjoint-paths_k[(x_1,y_1),…,(x_k,y_k)] that evaluate to true if there are internally vertex-disjoint paths between (the valuations of) x_i and y_i for all 1 ≤ i ≤ k. Disjoint-paths logic can express the disjoint paths problem, the problem of (topological) minor containment, the problem of hitting (topological) minors, and many more. Again we show that the fragments FO+DP_k that use predicates for at most k disjoint paths form a strict hierarchy of expressiveness. Finally, we compare the expressive power of the new logics with that of transitive-closure logics and monadic second-order logic.

We introduce a new data structure for answering connectivity queries in undirected graphs subject to batched vertex failures. Precisely, given any graph G and integer parameter k, we can in fixed-parameter time construct a data structure that can later be used to answer queries of the form: "are vertices s and t connected via a path that avoids vertices u₁,…, u_k?" in time 2^𝒪(k). In the terminology of the literature on data structures, this gives the first deterministic data structure for connectivity under vertex failures where for every fixed number of failures, all operations can be performed in constant time. With the aim to understand the power and the limitations of our new techniques, we prove an algorithmic meta theorem for the recently introduced separator logic, which extends first-order logic with atoms for connectivity under vertex failures. We prove that the model-checking problem for separator logic is fixed-parameter tractable on every class of graphs that exclude a fixed topological minor. We also show a weak converse. This implies that from the point of view of parameterized complexity, under standard complexity theoretical assumptions, the frontier of tractability of separator logic is almost exactly delimited by classes excluding a fixed topological minor. The backbone of our proof relies on a decomposition theorem of Cygan, Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh [SICOMP '19], which provides a tree decomposition of a given graph into bags that are unbreakable. Crucially, unbreakability allows to reduce separator logic to plain first-order logic within each bag individually. Guided by this observation, we design our model-checking algorithm using dynamic programming over the tree decomposition, where the transition at each bag amounts to running a suitable model-checking subprocedure for plain first-order logic. This approach is robust enough to provide also an extension to efficient enumeration of answers to a query expressed in separator logic.