We explore the interplay among three classical notions in graph theory: edge-colorings, perfect matchings, and graph coverings (locally bijective homomorphisms of graphs). In this paper, we consider undirected graphs in full generality of this notion: in contrast to the standard notion of a simple graph, our graphs may contain loops, semi-edges, and multiple edges. Many well-studied graph concepts, including matchings, edge-colorings, and covering projections, extend naturally to such graphs. Nevertheless, the role of simple graphs for graph covering problems is central, as emphasized in [J. Bok, J. Fiala, N. Jedličková, J. Kratochvíl, and M. Seifrtová. Computational complexity of covering disconnected multigraphs. Discret. Appl. Math., 359:229–243, 2024]. In that work, a relation "being stronger" was defined (a graph A is stronger than a graph B if every simple graph that covers A also covers B), and it was conjectured that if A has no semi-edges, then A is stronger than B if and only if A covers B. In their extended abstract presented at Eurocomb'23, Kratochvíl and Nedela proved this conjecture for 3-regular 1-vertex graphs B (and arbitrary A). They also introduced the notion (A,B)-snark for a simple graph G that demonstrates that A is not stronger than B. We continue this line of research in the current paper. As the main result, we show that for every graph A, there exists a simple graph D that covers A in such a way that the maximum number of pairwise disjoint perfect matchings equals the maximum number of pairwise disjoint perfect semi-matchings in A, i.e., spanning 1-regular subgraphs. Notably, the proof is constructive. As a corollary, we obtain a necessary condition for A to be stronger than B in general. This condition turns out to be sufficient whenever B is a 1-vertex graph (there are infinitely many of them), which, in particular, proves the aforementioned conjecture of Bok et al. in this case. Finally, we provide a constructive alternative to the existential NP-hardness proof of covering disconnected graphs in Bok et al. for the case when the target graph contains a 1-vertex component which itself determines an NP-hard covering problem.

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.

An outer-string representation of a graph is an intersection representation in which each vertex is represented by a curve that is contained in the unit disk and has at least one endpoint on the boundary of the unit disk. In an outer-1-string representation the curves representing any two vertices are in addition allowed to intersect at most once. In this paper, we consider the following constrained version: Given a graph G plus a cyclic order v1, . . . , vn of the vertices in G, test whether G has an outer-string or an outer-1-string representation in which the curves representing v1, . . . , vn intersect the boundary of the unit disk in this order. We first show that a graph has an outer-string representation for all possible cyclic orders of the vertices if and only if the graph is the complement of a chordal graph. Then we turn towards the situation where one particular cyclic order of the vertices is fixed. We characterize the chordal graphs admitting a constrained outer-string representation and the trees and cycles admitting a constrained outer-1-string representation. The characterizations yield polynomial-time recognition and construction algorithms; in the case of outer-1-string representations the run time is linear. We also show how to decide in polynomial time whether an arbitrary graph admits a constrained L-shaped outer-1-string representation. In an L-shaped representation the curves are 1-bend orthogonal polylines anchored on a horizontal line, and they are contained in the half-plane below that line. However, not even all paths with a constrained outer-1-string representation admit one with L-shapes. We show that 2-bend orthogonal polylines are sufficient for trees and cycles with a constrained outer-1-string representation.

Matchstick graphs are graphs that allow plane embedding with straight edges of equal length. One-planar unit distance graphs are graphs that allow a drawing in the plane in which all edges are straight-line segments of equal length and every edge crosses at most one other edge. The maximum number of edges of a matchstick graph (1-planar unit distance graph) of order n is denoted by u₀(n) (u₁(n), respectively). It is known that u₀(n) = ⌊ 3n-√{12n-3}⌋ holds for every n. At GD'24, Gehér and Tóth proved a slightly weaker upper bound on u₁(n), but noted that no 1-planar unit distance graph G with more than u₀(|V(G)|) vertices was known. They asked if u₁(n) = u₀(n) holds for every n. We give a negative answer to this question in a much stronger way. We show that u₁(n) > u₀(n) for every n ≥ 16135. Furthermore, we show that the gap between u₁(n) and u₀(n) can be arbitrarily large by proving that for n large enough with respect to a constant α < ∜{1/3}, u₁(n)-u₀(n) ≥ α∜{n}.

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.

We study a model of machine teaching where the teacher mapping is constructed from a size function on both concepts and examples. The main question in machine teaching is the minimum number of examples needed for any concept, the so-called teaching dimension. A recent paper (Ferri et al., 2024) conjectured that the worst case for this model, as a function of the size of the concept class, occurs when the consistency matrix contains the binary representations of numbers from zero and up. In this paper we prove their conjecture. The result can be seen as a generalization of a theorem resolving the edge isoperimetry problem for hypercubes (Hart, 1976), and our proof is based on a lemma of (Graham, 1970).

In 1992 Biró, Hujter and Tuza introduced, for every fixed connected graph $H$, the class of $H$-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of $H$. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of $H$-graphs for different graphs $H$. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, Töpfer, Voborn\'ık, and Zeman showed, for every fixed tree $T$, a polynomial-time algorithm recognizing $T$-graphs. Tucker showed a polynomial time algorithm recognizing $K_3$-graphs (circular-arc graphs). On the other hand, Chaplick at al. showed that recognition of $H$-graphs is $NP$-hard if $H$ contains two different cycles sharing an edge. The main two results of this work narrow the gap between the $NP$-hard and $P$ cases of $H$-graphs recognition. First, we show that recognition of $H$-graphs is $NP$-hard when $H$ contains two different cycles. On the other hand, we show a polynomial-time algorithm recognizing $L$-graphs, where $L$ is a graph containing a cycle and an edge attached to it ($L$-graphs are called lollipop graphs). Our work leaves open the recognition problems of $M$-graphs for every unicyclic graph $M$ different from a cycle and a lollipop. Other results of this work, which shed some light on the cases that remain open, are as follows. Firstly, the recognition of $M$-graphs, where $M$ is a fixed unicyclic graph, admits a polynomial time algorithm if we restrict the input to graphs containing particular holes (hence recognition of $M$-graphs is probably most difficult for chordal graphs). Secondly, the recognition of medusa graphs, which are defined as the union of $M$-graphs, where $M$ runs over all unicyclic graphs, is $NP$-complete.

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).

Abstract Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. 10.1016/j.ipl.2017.01.007 ). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs - a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

We study the following problem: given a planar graph G and a planar drawing (embedding) of a subgraph of G , can such a drawing be extended to a planar drawing of the entire graph G ? This problem fits the paradigm of extending a partial solution for a problem to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes an otherwise easy problem hard, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmas, which show that the planarity of partially embedded graphs exhibits the ‘TONCAS’ behavior “the obvious necessary conditions for planarity are also sufficient.” These conditions are expressed in terms of the interplay between (1) the rotation system and containment relationships between cycles and (2) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we make our algorithm run in linear time. Finally, we consider several generalizations of the problem, such as minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. We also apply our algorithm to the simultaneous graph drawing problem Simultaneous Embedding with Fixed Edges (Sefe) . There we obtain a linear-time algorithm for the case that one of the input graphs or the common graph has a fixed planar embedding.

A partially embedded graph (or PEG) is a triple (G,H,EH), where G is a graph, H is a subgraph of G, and EH is a planar embedding of H. We say that a PEG (G,H,EH) is planar if the graph G has a planar embedding that extends the embedding EH.

Geometric representations of graphs are intensively studied for their practical motivations and applications, as well as interesting structural and theoretical properties. We will survey recent results, persistent open problems, and prospective directions of further research in this exciting area lying on the border of graph theory and computational geometry.

We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “on-cas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach linear-time complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.

A straight-line drawing δ of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G,δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-drawing problem. Further we define fix(G,δ)=n−shift(G,δ) to be the maximum number of vertices of a planar n-vertex graph G that can be fixed when untangling δ. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log\log n}$ vertices when untangling a drawing of an n-vertex graph G. If G is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$ vertices. On the other hand, we construct, for arbitrarily large n, an n-vertex planar graph G and a drawing δ G of G with $\ensuremath {\mathrm {fix}}(G,\delta_{G})\leq \sqrt{n-2}+1$ and an n-vertex outerplanar graph H and a drawing δ H of H with $\ensuremath {\mathrm {fix}}(H,\delta_{H})\leq2\sqrt{n-1}+1$ . Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

A ranking of a graph is a coloring of the vertex set with positive integers in such a way that on every path connecting two vertices of the same color there is a vertex of larger color. We consider the directed variant of this problem, where the above condition is imposed only on those paths in which all edges are oriented consecutively. We show that the ranking number of an orientation of a tree is bounded by that of its longest directed path plus one, and that it can be computed in polynomial time. Unlike the undirected case, however, deciding whether the ranking number of a directed (and even of an acyclic directed) graph is bounded by a constant is NP-complete. In fact, the 3-ranking of planar bipartite acyclic digraphs is already hard.

A Boolean formula in a conjunctive normal form is called a $(k,s)$ – formula if every clause contains exactly k variables and every variable occurs in at most s clauses. The $(k,s)$–${\text{SAT}}$ problem is the SATISFIABILITY problem restricted to $(k,s)$–formulas. It is proved that for every $k \geqslant 3$ there is an integer $f(k)$ such that $(k,s)$–${\text{SAT}}$ is trivial for $s \leqslant f(k)$ (because every $(k,s)$–formula is satisfiable) and is NP-complete for $s \geqslant f(k) + 1$. Moreover, $f(k)$ grows exponentially with k, namely, $\lfloor {{{2^k } / {ek}}} \rfloor \leqslant f(k) \leqslant 2^{k - 1} - 2^{k - 4} - 1$ for $k \geqslant 4$.

The computational complexity of the following type of problems is studied. Given a topological layout (i.e., a drawing in the plane) of a graph, does it contain a noncrossing subgraph of a given type? It is conjectured that such problems are always NP-hard (provided planar subgraphs are looked for) regardless of the complexity of their nonplanar versions. This conjecture is verified for several cases in a very strong sense. In particular, it is shown that deciding the existence of a noncrossing path connecting two given vertices in a given topological layout of a 3-regular subgraph, as well as deciding the existence of a noncrossing cycle in such a layout, are NP-complete problems. It is also proved that deciding the existence of a noncrossing k-factor in a topological layout of a $( k + 1 )$-regular graph is NP-complete for $k = 2,3,4,5$. For $k = 1$, this question is NP-complete in layouts of 3-regular graphs, while it is polynomial solvable for layouts of graphs with maximum degree two.

A two-factor of G consists of disjoint cycles that cover $V( G )$. The authors consider the existence problem for two-factors in which the cycles are restricted to having lengths from a prescribed (possibly infinite) set of integers. Theorems are presented which derive the existence of such restricted two-factors in G from their existence in $G - u$ and $G - v $. The possibility of such theorems is then related to the complexity of the corresponding existence problem. In particular, the only four cases in which polynomial algorithms can be expected (in the sense that all other cases are shown to be NP-hard) are identified.