The classic theorem of Gallai and Milgram (1960) generalizes several fundamental results in Graph Theory, such as Dilworth’s theorem on posets and Kőnig’s theorem on matchings in bipartite graphs. The theorem asserts that for every graph G, the vertex set of G can be partitioned into at most α(G) vertex-disjoint paths, where α(G) is the maximum size of an independent set in G. The proof of the Gallai-Milgram theorem is constructive and yields a polynomial-time algorithm that computes a covering of G by at most α(G) vertex-disjoint paths. While the Gallai-Milgram theorem is tight—there are graphs where one really needs α(G) paths, not fewer, to cover the vertex set of G—it was not known prior to our work whether deciding if a graph G could be covered by fewer than α(G) vertex-disjoint paths can be done in polynomial time. We resolve this question by proving the following algorithmic extension of the Gallai–Milgram theorem for undirected graphs: There is an algorithm that, for an n-vertex graph G and an integer parameter k ≥ 1, runs in time 22O(k4logk) · nO(1) and outputs a path cover P of G together with either a correct conclusion that P is a minimum-size path cover or an independent set of size |P| + k, certifying that P contains at most α(G) − k paths. Thus, for k ∈ O((loglogn)1/4−ε) our algorithm runs in polynomial time, and either computes a minimum-size path cover of G, or finds a path cover of size at most α(G) − k. We find the existence of such an algorithm quite surprising for the following reason. The problems of computing a path cover and a maximum independent set are both notoriously hard, yet our algorithm either solves one of them or provides meaningful information about the other. The proof of our algorithmic extension of the Gallai–Milgram theorem is non-trivial and builds on several novel algorithmic ideas. One of the key subroutines in our algorithm is an FPT algorithm, parameterized by α(G), for deciding whether G contains a Hamiltonian path. This result is of independent interest—prior to our work, no polynomial-time algorithm for deciding Hamiltonicity was known, even for graphs with independence number at most three. Moreover, the algorithmic techniques we develop apply to a wide array of problems in undirected graphs, including Hamiltonian Cycle, Path Cover, Largest Linkage, and Topological Minor Containment. We show that all these problems are FPT when parameterized by the independence number of the graph. Notably, the independence-number parameterization departs from the typical direction of research in parameterized complexity. First, α(G) measures a graph’s density, whereas most prior work in the area focuses on parameters describing sparsity, such as treewidth or vertex cover. Second, most structural parameters studied in parameterized complexity can be computed exactly or well-approximated in polynomial or even FPT time, whereas computing α(G) is notoriously difficult from almost any computational perspective. The fact that it can nevertheless serve as the basis for efficient parameterization is particularly striking.

A graph covering projection, also referred to as a locally bijective homomorphism, is a mapping between the vertices and edges of two graphs that preserves incidences and is a local bijection. This concept originates in topological graph theory but has also found applications in combinatorics and theoretical computer science. In this paper we consider undirected graphs in the most general setting - graphs may contain multiple edges, loops, and semi-edges. This is in line with recent trends in topological graph theory and mathematical physics. We advance the study of the computational complexity of the H-Cover problem, which asks whether an input graph allows a covering projection onto a parameter graph H. The quest for a complete characterization started in 1990’s. Several results for simple graphs or graphs without semi-edges have been known, the role of semi-edges in the complexity setting has started to be investigated only recently. One of the most general known NP-hardness results states that H-Cover is NP-complete for every simple connected regular graph of valency greater than two. We complement this result by considering regular graphs H arising from connected acyclic graphs by adding semi-edges. Namely, we prove that any graph obtained by adding semi-edges to the vertices of a tree making it a d-regular graph with d ≥ 3, defines an NP-complete graph covering problem. In line with the so called Strong Dichotomy Conjecture, we prove that the NP-hardness holds even for simple graphs on input.

A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring , Star Colouring and Injective Colouring . We give almost complete complexity classifications for Acyclic Colouring , Star Colouring and Injective Colouring on H -free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours k is fixed, that is, not part of the input. From our study it follows that for fixed k , the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.

We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for graphs with semi-edges. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello et al. asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show that the presence of semi-edges makes the covering problem considerably harder; e.g., it is no longer sufficient to specify the vertex mapping induced by the covering, but one necessarily has to deal with the edge mapping as well. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding k-tuples of pairwise disjoint perfect matchings in regular graphs, or finding equitable partitions of regular bipartite graphs).

A k-colouring c of a graph G is a mapping V(G) → {1,2,… k} such that c(u) ≠ c(v) whenever u and v are adjacent. The corresponding decision problem is Colouring. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1,1)-Labelling). A classical complexity result on Colouring is a well-known dichotomy for H-free graphs, which was established twenty years ago (in this context, a graph is H-free if and only if it does not contain H as an induced subgraph). Moreover, this result has led to a large collection of results, which helped us to better understand the complexity of Colouring. In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We initiate such a systematic complexity study, and similar to the study of Colouring we use the class of H-free graphs as a testbed. We prove the following results: 1) We give almost complete classifications for the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring for H-free graphs. 2) If the number of colours k is fixed, that is, not part of the input, we give full complexity classifications for each of the three problems for H-free graphs. From our study we conclude that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs.

A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of closed walks as one of the key structural properties of a signed graph, we define a homomorphism of a signed graph $(G,σ)$ to a signed graph $(H, π)$ to be a mapping of vertices and edges of $G$ to (respectively) vertices and edges of $H$ which preserves incidence, adjacency and the signs of closed walks. In this work we first give a characterization of the sets of closed walks in a graph $G$ that correspond to the set of negative walks in some signed graph on $G$. We also give an easy algorithm for the corresponding decision problem. After verifying the equivalence between this definition and earlier ones, we discuss the relation between homomorphisms of signed graphs and those of 2-edge-colored graphs. Next we provide some basic no-homomorphism lemmas. These lemmas lead to a general method of defining chromatic number which is discussed at length. Finally, we list a few problems that are the driving force behind the study of homomorphisms of signed graphs.