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In the Choosability problem (or list chromatic number problem), for a given graph G, we need to find the smallest k such that G admits a list coloring for any list assignment where all lists contain at least k colors. The problem is tightly connected with the well-studied Coloring and List Coloring problems. However, the knowledge of the complexity landscape for the Choosability problem is pretty scarce. Moreover, most of the known results only provide lower bounds for its computational complexity and do not provide ways to cope with the intractability. The main objective of our paper is to construct the first non-trivial exact exponential algorithms for the Choosability problem, and complete the picture with parameterized results. Specifically, we present the first single-exponential algorithm for the decision version of the problem with fixed k. This result answers an implicit question from Eppstein on a stackexchange thread discussing upper bounds on the union of lists assigned to vertices. We also present a (2^{n^2} poly(n)) time algorithm for the general Choosability problem. In the parameterized setting, we give a polynomial kernel for the problem parameterized by vertex cover, and algorithms that run in FPT time when parameterized by a size of a clique-modulator and by the dual parameterization (n-k). Additionally, we show that Choosability admits a significant running time improvement if it is parameterized by cutwidth in comparison with the parameterization by treewidth studied by Marx and Mitsou [ICALP’16]. On the negative side, we provide a lower bound parameterized by a size of a modulator to split graphs under assumption of the Exponential Time Hypothesis.

The distance energy of a simple undirected graph (mathcal {G}), denoted by (mathcal {E}_D(mathcal {G})), is the sum of the absolute values of the eigenvalues of the distance matrix (D(mathcal {G})) of (mathcal {G}). In this paper, we study the effects on distance energy of some special complete t-partite graphs due to embedding an edge. This paper is motivated by the study in a 2022 paper by Wang and Meng.

We extend the notion of the Dyck language from words to two-dimensional arrays of symbols, i.e., pictures, using the row-column combination (also known as the crossword) of two Dyck languages over the same alphabet. In a Dyck crossword picture, each column and each row must be a word from the respective Dyck language. The pairing of open and closed parentheses in a Dyck word can be represented by edges connecting corresponding cells in the same row or column. This defines a matching graph, which serves as the two-dimensional analogue of the syntactic tree of a Dyck word. A matching graph is partitioned into simple circuits of unbounded length (always a multiple of four), whose labels form a regular language. These circuits exhibit a wide variety of forms and labelings, which we illustrate and partially classify. With a two-letter alphabet, a Dyck crossword is necessarily empty. The minimal non-trivial case, requiring an alphabet of size four, already generates all possible forms of matching graphs and is the primary focus of our study. We prove that the only picture with a single matching circuit (i.e., a Hamiltonian cycle) has size 2 by 2. Two key properties of Dyck words–cancellation and well-nesting–can be generalized to two dimensions, leading to two alternative definitions of 2D Dyck languages: neutralizable and well-nested. These languages are special cases of Dyck crossword pictures called quaternate, where all circuits have length 4 (i.e., are rectangles). This results in a strict language inclusion hierarchy: well-nested (subset ) neutralizable (subset ) quaternate (subset ) Dyck crosswords. When the alphabet size exceeds four, not all combinations of row and column Dyck languages yield non-empty crosswords. To identify productive combinations, we introduce an alphabetic graph, where nodes represent alphabet symbols and edges represent their couplings. A matching circuit corresponds to the unrolling of an alphabetic graph circuit. Finally, we prove that Dyck crosswords are not tiling-recognizable, as expected for a definition extending Dyck word languages to pictures.

We study deterministic tree-walking-storage automata, which are finite-state devices equipped with a tree-like storage. These automata are generalized stack automata, where the linear stack storage is replaced by a non-linear tree-like stack. Therefore, tree-walking-storage automata have the ability to explore the interior of the tree storage without altering the contents, with the possible moves of the tree pointer corresponding to those of tree-walking automata. In addition, a tree-walking-storage automaton can append (push) non-existent descendants to a tree node and remove (pop) leaves from the tree. Here we are particularly considering the capacities of deterministic tree-walking-storage automata working in real time. It is shown that even the non-erasing variant can accept rather complicated unary languages as, for example, the language of words whose lengths are powers of two, or the language of words whose lengths are double Fibonacci numbers. Comparing the computational capacities with automata from the classical automata hierarchy, we derive that the family of languages accepted by real-time deterministic (non-erasing) tree-walking-storage automata is located between the regular and the deterministic context-sensitive languages. Moreover, the families are incomparable with the families of context-free and growing context-sensitive languages. It turns out that the devices under consideration accept unary languages in non-erasing mode that cannot be accepted by any classical stack automaton, even in erasing mode and arbitrary time. Basic closure properties of the induced families of languages are shown. In particular, we consider Boolean operations and AFL operations. It turns out that the two families in question have the same properties and, in particular, share all but one of these closure properties with the important family of deterministic context-free languages. Then, we consider the computational capacity of the counterpart to counter- and stack-counter automata, where the set of stack symbols is a singleton. Finally, we explore several decidability problems and show, that even for devices with a single tree symbol, the problems are all non-semidecidable by reductions of non-semidecidable problems of Turing machines.

Does every three-word code have a finite completion? Up to now, this famous question in the theory of codes remains open. Motivated by this problem, we construct several types of three-word codes with the form ({a, aba, u}) and ({a, ab, v}) which have finite completions.

Bipartite graphs are commonly used to visualize objects and their features. An object may possess several features and several objects may share a common feature. The standard visualization of bipartite graphs, with objects and features on two (say horizontal) parallel lines at integer coordinates and edges drawn as line segments, can often be difficult to work with. A common task in visualization of such graphs is to consider one object and all its features. This naturally defines a drawing window, defined as the smallest interval that contains the x-coordinates of the object and all its features. We show that if both objects and features can be reordered, minimizing the average window size is NP-hard. However, if the features are fixed, then we provide an efficient polynomial-time algorithm for arranging the objects, so as to minimize the average window size. Finally, we introduce a different way of visualizing the bipartite graph, by placing the nodes of the two parts on two concentric circles. For this setting we also show NP-hardness for the general case and a polynomial-time algorithm when the features are fixed.

In the thief orienteering problem an agent called a thief carries a knapsack of capacity W and has a time limit T to collect a set of items of total weight at most W and maximum profit along a simple path in a weighted graph (G = (V, E)) from a start vertex s to an end vertex t. There is a set I of items each with weight (w_{i}) and profit (p_{i}) that are distributed among (V{setminus }{s,t}). The time needed by the thief to travel an edge depends on the length of the edge and the weight of the items in the knapsack at the moment when the edge is traversed. There is a polynomial-time approximation scheme for a relaxed version of the thief orienteering problem on directed acyclic graphs that produces solutions that use time at most (T(1 + epsilon )) for any constant (epsilon > 0). We give a polynomial-time algorithm for transforming instances of the problem on 2-terminal series–parallel graphs into equivalent instances of the thief orienteering problem on directed acyclic graphs; therefore, yielding a polynomial-time approximation scheme for the relaxed version of the thief orienteering problem on this graph class.

A graph (G=(V,E)) is a star-k-pairwise compatibility graph (star-k-PCG) if there exists a weight function (w: V rightarrow mathbb {R}^+) and k mutually exclusive intervals (I_1, I_2, ldots I_k), such that there is an edge (uv in E) if and only if (w(u)+w(v) in bigcup _i I_i). These graphs are related to two important classes of graphs: pairwise compatibility graphs (PCGs) and multithreshold graphs. It is known that for any graph G there exists a k such that G is a star-k-PCG. Thus, for a given graph G it is interesting to know which is the minimum k such that G is a star-k-PCG. We define this minimum k as the star number of the graph, denoted by (gamma (G)). Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of (gamma (G)) for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two-dimensional grid graphs is 2 and that the star number of 4-dimensional grids is at least 3. Finally, we conclude with numerous open problems.

For (tge 3), (K_{1, t}) is called t-claw. A graph (G=(V, E)) is t-claw free if it does not contain t-claw as a vertex-induced subgraph. In minimum t-claw deletion problem (Min-t-Claw-Del), given a graph (G=(V, E)), it is required to find a vertex set S of minimum size such that (G[Vsetminus S]) is t-claw free. In a split graph, the vertex set is partitioned into two sets such that one forms a clique and the other forms an independent set. Every t-claw in a split graph has a center vertex in the clique partition. This observation motivates us to consider the minimum one-sided bipartite t-claw deletion problem (Min-t-OSBCD). Given a bipartite graph (G=(A cup B, E)), in Min-t-OSBCD it is asked to find a vertex set S of minimum size such that (G[(A cup B) {setminus } S]) has no t-claw with the center vertex in A. A primal-dual algorithm approximates Min-t-OSBCD within a factor of t. We prove that it is ({textsf{UGC}})-hard to approximate with a factor better than t. We also prove it is approximable within a factor of 2 for dense bipartite graphs. By using these results on Min-t-OSBCD, we prove that Min-t-Claw-Del is ({textsf{UGC}})-hard to approximate within a factor better than t, for split graphs. We also consider their complementary maximization problems and prove that they are ({textsf{APX}})-complete.

The piecewise complexity h(u) of a word is the minimal length of subwords needed to exactly characterise u. Its piecewise minimality index (rho (u)) is the smallest length k such that u is minimal among its order-k class ([u]_k) in Simon’s congruence. We initiate a study of these two descriptive complexity measures. Among other results, we provide efficient algorithms for computing h(u) and (rho (u)) for a given word u.