The study of word-representability was initiated by the seminal work of Kitaev and Pyatkin in 2008 that has later led to the monograph by Kitaev and Lozin in 2015. In this paper, we build on the very recent work by Fernau et al. who proposed a general framework that generalizes certain aspects of word-representability, so that any binary language describes a graph class. In this work, we systematically study particularly small languages and observe that they characterize well-known graph classes, e.g., interval, permutation, circle, and bipartite chain graphs. Thus, we strengthen the bond between formal languages and graph classes, even for small binary languages. We also show some limitations of our approach by proving that, e.g., families of sparse graphs like planar graphs cannot be characterized by any language following this approach.

In this paper, we address the enumeration of (induced) s-t paths and minimal s-t separators. These problems are some of the most famous classical enumeration problems that can be solved in polynomial delay by simple backtracking for a (un)directed graph. As a generalization of these problems, we consider the (induced) s-t hyperpath and minimal s-t separator enumeration in a directed hypergraph. We show that extending these classical enumeration problems to directed hypergraphs drastically changes their complexity. More precisely, there are no output-polynomial time algorithms for the enumeration of induced s-t hyperpaths and minimal s-t separators unless = , and the s-t hyperpath enumeration is at least as hard as the minimal transversal enumeration, even if a directed hypergraph is BF-hypergraph. As a positive result, s-t hyperpath enumeration for a B-hypergraph can be solved in polynomial delay by backtracking.

In this work, we introduce a new notion for representing graph classes with formal languages. In contrast to the seminal work by Kitaev and Pyatkin to represent graphs by words, we use formal binary languages in order to have a set of patterns (given by the languages' words) defining the edges in the graph. In particular, we investigate famous languages like the palindromes, copy-words, Lyndon words, and Dyck words to represent all graphs or specific graph classes by restricting these languages.

In this paper, we study the task of enumerating (and counting) locally and globally minimal defensive alliances in graphs. We consider general graphs as well as special graph classes. From an input-sensitive perspective, our presented algorithms are mostly optimal.

We initiate the study of the shortest reconfiguration problem for independent sets under the adjacency relation derived from the independent set polytope. Given a graph and two independent sets, the problem asks for a shortest sequence transforming one into the other such that the subgraph induced by the symmetric difference of any two consecutive sets is connected. This is equivalent to finding a shortest path on the 1-skeleton of the independent set polytope. We prove that the problem is NP-hard even on planar graphs of bounded degree, as well as on split graphs. Notably, the hardness for planar graphs of bounded degree still holds even when deciding whether the target can be reached in at most two steps. For split graphs, we further show the W[2]-hardness when parameterized by the number of steps, as well as the inapproximability of the optimal length. As a consequence, we prove that the length of a shortest path between two vertices of a 0/1 polytope in ℝ^n described by O(n) linear inequalities is hard to approximate within a factor of (1-ε)ln n for any constant ε>0, unless P=NP. On the positive side, we provide polynomial-time algorithms for block graphs, cographs, and bipartite chain graphs. Moreover, for paths and cycles, we show that the optimal length of the shortest reconfiguration sequence exactly matches a trivial upper bound.

Different variations of alliances in graphs have been introduced into the graph-theoretic literature about twenty years ago. More broadly speaking, they can be interpreted as groups that collaborate to achieve a common goal, for instance, defending themselves against possible attacks from outside. In this paper, we initiate the study of reconfiguring alliances. This means that, with the understanding of having an interconnection map given by a graph, we look at two alliances of the same size k and investigate if there is a reconfiguration sequence (of length at most ℓ ) formed by alliances of size (at most) k that transfers one alliance into the other one. Here, we consider different (now classical) movements of tokens: sliding, jumping, addition/removal. We link the latter two regimes by introducing the concept of reconfiguration monotonicity. Concerning classical complexity, most of these reconfiguration problems are PSPACE-complete, although some are solvable in LogSPACE. We also consider these reconfiguration questions through the lense of parameterized algorithms and prove various FPT-results, in particular concerning the combined parameter k+ℓ or neighborhood diversity together with k or neighborhood diversity together with ℓ .

Although Extension Perfect Roman Domination is NP-complete, all minimal (with respect to the pointwise order) perfect Roman dominating functions can be enumerated with polynomial delay. This algorithm uses a bijection between minimal perfect Roman dominating functions and Roman dominating functions and the fact that all minimal Roman dominating functions can be enumerated with polynomial delay. This bijection considers the set of vertices with value 2 under the functions. In this paper, we will generalize this idea by defining so called nice Roman Domination properties for which we can employ this method. With this idea, we can show that all minimal maximal Roman Dominating functions can be enumerated with polynomial delay in O(1.9332^n) time. Furthermore, we prove that enumerating all minimal connected/total Roman dominating functions on cobipartite graphs can be achieved with polynomial delay. Additionally, we show the existence of a polynomial-delay algorithm for enumerating all minimal connected Roman dominating function on interval graphs. We show some downsides to this method as well.

Roman domination is one of few examples where the related extension problem is polynomial-time solvable even if the original decision problem is NP-complete. This is interesting as it allows to establish polynomial-delay enumeration algorithms for finding minimal Roman dominating functions, while it is open for more than four decades if all minimal dominating sets can be enumerated efficiently. To find the reason why this is the case, we combine the idea of hitting set with the idea of Roman domination. We hence obtain a new combinatorial problem, called Roman Hitting Set, generalizing Roman Domination in a natural way. This allows us to expand the frontier of polynomial-delay enumerability, as opposed to transversal-hardness. The generalization Roman Hitting Set is insofar interesting as it always allows for polynomial-delay enumeration, and we also give an explicit input-sensitive enumeration algorithm for this problem that is optimal in the sense that we can also give a matching lower bound. Based on Roman Hitting Set, we also discuss consequences for Roman variants of Vertex Cover and Feedback Vertex Set. For minimal Roman vertex covers of size upper-bounded by k, we can also deduce an optimal parameterized enumeration algorithm. Also viewed from Parameterized Complexity, the studies on extension problems are interesting, giving more examples of parameterized problems complete for [3] . Finally, we also consider implicitly given hypergraphs and prove that, for the example of Roman FVS, our polynomial-delay enumeration algorithm can still be used, avoiding an explicit construction of the hypergraph.

For an introductory single lecture into parameterized complexity classes, it appears to be necessary to introduce different problems to capture the (most important) different levels of the W-hierarchy. This puts an additional burden on the audience, as they also have to understand the different problems and not only the different complexity classes. In this paper, we will show that Extension Perfect Roman Domination, Ext PRD for short, is a single problem that can be used for this introductory purpose. We can characterize the classes [1] , [2] and [3] and find problems in FPT, XP and those being para-NP-hard. Also, it can be used to explain how the choice of the parameter can influence the complexity status of the problem. One can even touch fine-grained complexity by establishing a run-time lower bound based on the k-Orthogonal Vector conjecture. Ext PRD, being a sibling to Roman Domination, also comes with a nice story that can be also seen from the perspective of adventure games and might hence be appealing.

We introduce a new graph parameter called generalized lettericity as a generalization of the lettericity parameter that was introduced two decades ago. Given a word w=w_1w_2⋯ w_n of length |w|=n over a finite alphabet and a decoder set 𝒟 of words over with length at least 2, we define a generalized letter graph G(𝒟,w) with respect to 𝒟 and w. G(𝒟,w) has vertex set [n] and there is an edge between i

The question to enumerate all (inclusionwise) minimal connected dominating sets in a graph of order n in time significantly less than 2n is an open question that was asked in many places. We answer this question affirmatively, by providing an enumeration algorithm that runs in time O(1.9896n), using polynomial space only. The key to this result is the consideration of this enumeration problem on 2-degenerate graphs, which is proven to be possible in time O(1.9767n). Apart from solving this old open question, we also show new lower bound results. More precisely, we construct a family of graphs of order n with \Omega(1.4890n) many minimal connected dominating sets, while previous examples achieved \Omega(1.4422n). Our example happens to yield 4-degenerate graphs. Additionally, we give lower bounds for the previously not considered classes of 2-degenerate and of 3-degenerate graphs, which are \Omega(1.3195n) and \Omega(1.4723n), respectively. We also address essential questions concerning output-sensitive enumeration. Namely, we give reasons why our algorithm cannot be turned into an enumeration algorithm that guarantees polynomial delay without much effort. More precisely, we prove that it is NP-complete to decide, given a graph G and a vertex set U, if there exists a minimal connected dominating set D with U subset of D, even if G is known to be 2-degenerate. Our reduction also shows that even any subexponential delay is not easy to achieve for enumerating minimal connected dominating sets. Another reduction shows that no FPT-algorithms can be expected for this extension problem concerning minimal connected dominating sets, parameterized by U. This also adds one more problem to the still rather few natural parameterized problems that are complete for the parameterized complexity class W[3]. We also relate our enumeration problem to the famous HITTING SET TRANSVERSAL problem, a problem open for more than four decades, which can be phrased in our context as the question to enumerate all minimal dominating sets of a graph with polynomial delay, by showing that a polynomial-delay enumeration algorithm for minimal connected dominating sets implies an affirmative algorithmic (polynomial-delay) solution to the HITTING SET TRANSVERSAL problem.

The analysis of social networks and community detection is a central theme in Artificial Intelligence. One line of research deals with finding groups of agents that could work together to achieve a certain goal. To this end, different notions of so-called clusters or communities have been introduced in the literature of graphs and networks. Among these, a defensive alliance is a kind of quantitative group structure. However, all studies on alliances so far have ignored one aspect that is central to the formation of alliances on a very intuitive level, assuming that the agents are preconditioned concerning their attitude towards other agents: they prefer to be in some group (or in an alliance) together with the agents they like, so that they are happy to help each other towards their common aim, possibly then working against the agents outside of their group that they dislike. Signed networks were introduced in the psychology literature to model liking and disliking between agents, generalizing graphs in a natural way. Hence, we propose the novel notion of a defensive alliance in the context of signed networks. We then investigate several natural algorithmic questions related to this notion. These, and also combinatorial findings, connect our notion to that of correlation clustering, which is a well-established idea of finding groups of agents within a signed network. Also, we introduce a new structural parameter for signed graphs, the signed neighborhood diversity snd, and exhibit a snd-parameterized algorithm that finds one of the smallest defensive alliances in a signed graph.

Roman domination formalizes a military strategy going back to Constantine the Great. Here, armies are placed in different regions. A region is secured if there is at least one army in this region orthere are two armies in one neighbored region. This simple strategy can be easily translated into a graph-theoretic question. The placement of armies is described by a function which maps each vertex to 0, 1 or 2. Such a function is called Roman dominating if each vertex with value 0 has a neighbor with value 2. Roman domination is one of few examples where the related (so-called) extension problem is polynomial-time solvable even if the original decision problem is NP-complete. This is interesting as it allows to establish polynomial-delay enumeration algorithms for listing minimal Roman dominating functions, while it is open for more than four decades if all minimal dominating sets of a graph or (equivalently) if all hitting sets of a hypergraph can be enumerated with polynomial delay, or even in output-polynomial time. To find the reason why this is the case, we combine the idea of hitting set with the idea of Roman domination. We hence obtain and study a new problem, called Roman Hitting Function, generalizing Roman Domination towards hypergraphs. This allows us to delineate the frontier of polynomial-delay enumerability. Our main focus is on the extension version of this problem, as this was the key problem when coping with Roman domination functions. While doing this, we find some conditions under which the Extension Roman Hitting Function problem is NP-complete. We then use parameterized complexity as a tool to get a better understanding of why Extension Roman Hitting Function behaves in this way. From an alternative perspective, we can say that we use the idea of parameterization to study the question what makes certain enumeration problems that difficult. Also, we discuss another generalization of Extension Roman Domination, where both a lower and an upper bound on the sought minimal Roman domination function is provided. The additional upper bound makes the problem hard (again), and the applied parameterized complexity analysis (only) provides hardness results. Also from the viewpoint of Parameterized Complexity, the studies on extension problems are quite interesting as they provide more and more examples of parameterized problems complete for W[3], a complexity class with only very few natural members known five years ago.

The literature on word-representable graphs is quite rich, and a number of variations of the original definition have been proposed over the years. We are initiating a systematic study of such variations based on formal languages. In our framework, we can associate a graph class to each language over the binary alphabet {0,1}. All graph classes that are language-representable in this sense are hereditary and enjoy further common properties. Besides word-representable graphs and, more generally, 1^k- or k-11-representable graphs, we can identify many more graph classes in our framework, like (co)bipartite graphs, (co)comparability graphs, to name a few. It was already known that any graph is 111- or 2-11-representable. When such representations are considered for storing graphs, 111- or 2-11-representability bears the disadvantage of being significantly inferior to standard adjacency matrices or lists. We prove that quite famous languages like the palindromes, the copy language or the Lyndon words can match the efficiency of standard graph representations. The perspective of language theory allows us to prove general results that hold for all graph classes that can be defined in this way. This includes certain closure properties (e.g., all language-definable graph classes are hereditary) as well as certain limitations (e.g., all language-representable graph classes contain graphs of arbitrarily large treewidth and of arbitrarily large degeneracy, except a trivial case). As each language describes a graph class, we can also ask decidability questions concerning graph classes, given a concrete presentation of a formal language. We also present a systematic study of graph classes that can be represented by languages in which each letter occurs at most twice. Here, we find graph classes like interval, permutation, circle, bipartite chain, convex, and threshold graphs.

A graph G is well-covered if every minimal vertex cover of G is minimum, and it is well-dominated if every minimal dominating set of G is minimum. Studies on well-covered graphs were initiated in [Plummer, JCT 1970], and well-dominated graphs were first introduced in [Finbow, Hartnell and Nowakow, AC 1988]. Well-dominated graphs are well-covered, and both classes have been widely studied in the literature. The recognition of well-covered graphs has been proved to be coNP-complete by [Chvátal and Slater, AODM 1993] and [Sankaranarayana and Stewart, Networks 1992], but the complexity of recognizing well-dominated graphs has been left open since their introduction. We close this complexity gap by proving that recognizing well-dominated graphs is coNP-complete. This solves a well-known open question (cf. [Levit and Tankus, DM 2017] and [Gözüpek, Hujdurovic and Milanič, DMTCS 2017]), which was first asked in [Caro, Sebő and Tarsi, JAlg 1996]. Although the problem has been open for a long time, our proof is surprisingly simple. Finally, we show that recognizing well-totally-dominated graphs is coNP-complete, answering a question of [Bahadır, Ekim, and Gözüpek, AMC 2021].

Signed graphs have been introduced to enrich graph structures expressing relationships between persons or general social entities, introducing edge signs to reflect the nature of the relationship, e.g., friendship or enmity. Independently, offensive alliances have been defined and studied for undirected, unsigned graphs. We join both lines of research and define offensive alliances in signed graphs, hence considering the nature of relationships. Apart from some combinatorial results, mainly on k-balanced and k-anti-balanced signed graphs (where the latter is a newly introduced family of signed graphs), we focus on the algorithmic complexity of finding smallest offensive alliances, looking at a number of parameterizations. While the parameter solution size leads to an FPT result for unsigned graphs, we obtain W[2]-completeness for the signed setting. We introduce new parameters for signed graphs, e.g., distance to weakly balanced signed graphs, that could be of independent interest. We show that these parameters yield FPT results. Here, we make use of the recently introduced parameter neighborhood diversity for signed graphs.

Perfect Roman Dominating Functions and Unique Response Roman Dominating Functions are two ways to translate perfect code into the framework of Roman Dominating Functions. We also consider the enumeration of minimal Perfect Roman Dominating Functions and show a tight relation to minimal Roman Dominating Functions. Furthermore, we consider the complexity of the underlying decision problems Perfect Roman Domination and Unique Response Roman Domination on special graph classes. For instance, split graphs are the first graph class for which Unique Response Roman Domination is polynomial-time solvable, while Perfect Roman Domination is NP-complete. Beyond this, we give polynomial-time algorithms for Perfect Roman Domination on interval graphs and for both decision problems on cobipartite graphs. However, both problems are NP-complete on chordal bipartite graphs. We show that both problems are W[1]-complete if parameterized by solution size and FPT if parameterized by the dual parameter or by clique width.

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We develop non-trivial enumeration algorithms for minimal Roman dominating functions with polynomial delay and polynomial space. Recall that the existence of a similar enumeration result for minimal dominating sets is open for decades. Our result is based on a polynomial-time algorithm for Extension Roman Domination: Given a graph G=(V,E) and a function f:V→{0,1,2} , is there a minimal Roman dominating function f̃ with f≤f̃ ? Here, ≤ lifts 0< 1< 2 pointwise; minimality is understood in this order. Our enumeration algorithm is also analyzed from an input-sensitive viewpoint, leading to a run-time estimate of 𝒪(1.9332^n) for graphs of order n; this is complemented by a lower bound example of Ω (1.7441^n) .

The idea of enumeration algorithms with polynomial delay is to polynomially bound the running time between any two subsequent solutions output by the enumeration algorithm. While it is open for more than four decades if all minimal dominating sets of a graph can be enumerated in output-polynomial time, it has recently been proven that pointwise-minimal Roman dominating functions can be enumerated even with polynomial delay. The idea of the enumeration algorithm was to use polynomial-time solvable extension problems. We use this as a motivation to prove that also two variants of Roman dominating functions studied in the literature, named perfect and unique response, can be enumerated with polynomial delay. This is interesting since Extension Perfect Roman Domination is W[1]-complete if parameterized by the weight of the given function and even W[2]-complete if parameterized by the number vertices assigned 0 in the pre-solution, as we prove. Otherwise, efficient solvability of extension problems and enumerability with polynomial delay tend to go hand-in-hand. We achieve our enumeration result by constructing a bijection to Roman dominating functions, where the corresponding extension problem is polynomimaltime solvable. Furthermore, we show that Unique Response Roman Domination is solvable in polynomial time on split graphs, while Perfect Roman Domination is NP-complete on this graph class, which proves that both variations, albeit coming with a very similar definition, do differ in some complexity aspects. This way, we also solve an open problem from the literature.

Roman domination is one of few examples where the related extension problem is polynomial-time solvable even if the original decision problem is NP-complete. This is interesting, as it allows to establish polynomial-delay enumeration algorithms for finding minimal Roman dominating functions, while it is open for more than four decades if all minimal dominating sets of a graph or if all hitting sets of a hypergraph can be enumerated with polynomial delay. To find the reason why this is the case, we combine the idea of hitting set with the idea of Roman domination. We hence obtain and study two new problems, called Roman Hitting Function and Roman Hitting Set, both generalizing Roman Domination. This allows us to delineate the borderline of polynomial-delay enumerability. Here, we assume what we call the Hitting Set Transversal Thesis, claiming that it is impossible to enumerate all minimal hitting sets of a hypergraph with polynomial delay. Our first focus is on the extension versions of these problems. While doing this, we find some conditions under which the Extension Roman Hitting Function problem is NP-complete. We then use parameterized complexity to get a better understanding of why Extension Roman Hitting Function behaves in this way. Furthermore, we analyze the parameterized and approximation complexity of the underlying optimization problems. We also discuss consequences for Roman variants of other problems like Vertex Cover.