We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a reset word) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. The class of strongly connected partial DFAs is precisely the class of DFAs recognizing the Kleene star of prefix codes. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case, this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs to the same problems for complete DFAs. In particular, this includes the Cerny and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other.
We prove an Erdős--Szekeres type result for finite words over $\mathbb{N}$ with repeated values. Specifically, we define a \emph{repeat} in a word to be an occurrence of a value which is not its first occurrence. We define an occurrence of a \emph{pattern} $π$ in a word $w$ to be a (not necessarily consecutive) subword of $w$ that is order isomorphic to $π$. In this note, we show that every word with $kn^6+1$ repeats contains one of the following patterns: $0^{k+2}$, $0011\cdots nn$, $nn\cdots1100$, $012 \cdots n012 \cdots n$, $012 \cdots nn\cdots 210$, $n\cdots 210012\cdots n$, $n\cdots 210n\cdots 210$. Moreover, when $k=1$, we show that this is best possible by constructing a word with $n^6$ repeats that does not contain any of these patterns. 11 pages, 9 figures
A stable or locally-optimal cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. In this paper we study Minimum Stable Cut, the problem of finding a stable cut of minimum weight. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We begin by showing that the problem remains weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this hardness with a pseudo-polynomial DP algorithm solving the problem in time $(Δ\cdot W)^{O(tw)}n^{O(1)}$, where $tw$ is the treewidth, $Δ$ the maximum degree, and $W$ the maximum weight. On the other hand, bounding $Δ$ is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Minimum Stable Cut by both $tw$ and $Δ$ and obtain an FPT algorithm running in time $2^{O(Δtw)}(n+\log W)^{O(1)}$. Our main result for the weighted problem is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in $(nW)^{o(pw)}$ or $2^{o(Δpw)}(n+\log W)^{O(1)}$, then the ETH is false. Complementing this, we show that we can, however, obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions, that is, solutions where no single vertex can unilaterally increase the weight of its incident cut edges by more than a factor of $(1+\varepsilon)$. Motivated by these mostly negative results, we consider Unweighted Minimum Stable Cut. Here our results already imply a much faster exact algorithm running in time $Δ^{O(tw)}n^{O(1)}$. We show that this is also probably essentially optimal: an algorithm running in $n^{o(pw)}$ would contradict the ETH. Full version of ICALP 2021 paper
A solid polycube is a face-connected set of unit cubes, where, unlike classical definitions for polycubes, shared faces between adjacent cubes are not removed. We prove by mathematical induction that any such polycube can be edge-unfolded into a 2D net without refinement. Our proof relies on the concept of a perfect net, defined as a net in which two designated edges are placed on the far left and far right sides. This configuration enables consistent gluing of nets in a single direction throughout the induction process, thereby guaranteeing that no overlaps occur at any step.
Final version updated according to the journal (DMTCS) requirements, including corrected affiliations and layout adjustments. An \textit{AVD-$k$-total coloring} of a simple graph $G$ is a mapping $\pi:V(G) \cup E(G) \to \{1,\ldots,k\}$, with $k \geq 1$ such that: for each pair of adjacent or incident elements $x,y \in V(G) \cup E(G)$, $\pi(x) \neq \pi(y)$; and for each pair of adjacent vertices $x,y \in V(G)$, sets $\{\pi(x)\} \cup \{\pi(xv) \mid xv \in E(G), v \in V(G)\}$ and $\{\pi(y)\} \cup \{\pi(yv)\mid yv \in E(G), v \in V(G)\}$ are distinct. The \textit{AVD-total chromatic number}, denoted by $\chi''_{a}(G)$ is the smallest $k$ for which $G$ admits an AVD-$k$-total-coloring. We consider a conjecture proposed in 2010 in the thesis of Jonathan Hulgan that any graph~$G$ with maximum vertex degree 3 has $\chi''_{a}(G) \leq 5$. As positive evidence, we prove that several molecular graphs known as fullerene graphs have AVD-total chromatic number equal to 5.
Let (X, E ) be a hypergraph. A support is a graph Q on X such that for each E E E, the subgraph of Q induced on the elements in E is connected. We consider hypergraphs defined by connected subgraphs of a host graph. For a graph G = (V, E), let b(V ) C_ V denote a set of terminals. Given a collection 7-t of connected subgraphs of G, we define a hypergraph on b(V ), where each H E 7-t defines a hyperedge V (H) n b(V ). Our goal is to construct a graph Q on b(V ) so that for each H E 7-t, V (H) n b(V ) induces a connected subgraph of Q. We also consider the problem of constructing a support for the dual hypergraph - a hypergraph on 7-t where each v E b(V ) defines a hyperedge consisting of the subgraphs in 7-t containing v. In fact, we construct supports for a common generalization of the primal and dual settings called the intersection hypergraph. As our main result, we show that if the host graph G has genus g and the subgraphs in 7-t satisfy a condition of being cross-free, then there exists a support that also has genus at most g. Our results are a generalization of the results of Raman and Ray (Rajiv Raman, Saurabh Ray: Constructing Planar Support for Non-Piercing Regions. Discret. Comput. Geom. 64(3): 1098-1122 (2020)) and our techniques extend their results from the planar setting to graphs on surfaces. In particular, our techniques imply a unified analysis for packing and covering problems for hypergraphs defined on surfaces of bounded genus. We also describe applications of our results for hypergraph colorings.
Given a digraph, an ordering of its vertices defines a backedge graph, namely the undirected graph whose edges correspond to the arcs pointing backwards with respect to the order. The degreewidth of a digraph is the minimum over all ordering of the maximum degree of the backedge graph. We answer an open question by Keeney and Lokshtanov [WG 2024], proving that it is NP-hard to determine whether an oriented graph has degreewidth at most 1, which settles the last open case for oriented graphs. We complement this result with a general discussion on parameters defined using backedge graphs and their relations to classical parameters.
An AVD-k-total coloring of a simple graph G is a mapping pi : V (G)UE(G) -> {1, ... , k}, with k >= 1 such that: for each pair of adjacent or incident elements x, y is an element of V (G) U E(G), pi(x) not equal pi(y); and for each pair of adjacent vertices x, y is an element of V (G), sets {pi(x)} U {pi(xv) xv is an element of E(G) and v is an element of V (G)} and {pi(y)} U {yv is an element of E(G) and v is an element of V (G)} are distinct. The AVD-total chromatic number, denoted by chi '' (a)(G) is the smallest k for which G admits an AVD-k-totalcoloring. In 2010, Hulgan conjectured that any graph G with maximum vertex degree 3 has chi '' (a)(G) <= 5. As positive evidence, we prove that several molecular graphs known as fullerene graphs have AVD-total chromatic number equal to 5.
In this paper, we consider induced subgraphs of the Hamming graph H(n, 3). We show that if U subset of Z(3)(n) and U induces a subgraph of H(n, 3) with maximum degree at most 1 then 1. If U is disjoint from a maximum size independent set of H(n, 3) then |U| <= 3(n-1) + 1. Moreover, all such U with size 3(n-1) + 1 are isomorphic to each other. 2. For n >= 6, there exists such a U with size |U| = 3(n-1) + 18 and this is optimal for n = 6. 3. If U boolean AND {x, x + e(1), x + 2(e1)} =/ phi for all x is an element of Z(3)(n) then |U| <= 3(n-1) + 81.
In an effort to further understanding q, t-Catalan statistics, a new statistic on Dyck paths called depth was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known area statistics. In a recent preprint, Qu and Zhang (2025) generalized depth to so-called "(k) over right arrow -Dyck paths". They showed that area and depth are also jointly equi-distributed over such paths with a fixed multiset of up-steps and a given first up-step, and they conjectured that the same holds when also fixing the last up-step. In this short note, we settle this conjecture on the more general context of Lukasiewicz paths by interpreting area and depth under the classical bijection between Lukasiewicz paths and plane trees, through which the symmetry is transparent.
Recently, in the context of walks of hexagonal circle packings, interest has emerged in the family of skew Dyck paths with two variants of down-steps. These paths have steps $U, D_g, D_b, L=D_r$. Using generating functions, the kernel method and (in)finite linear systems, contributions to the (average) height and other enumerations are made. As in many similar instances, the average height is of order $\sqrt n$.
In 1977, Chung, Chung and Liu generalized the definition of the Ramsey number. They introduced the s-chromatic Ramsey number as follows. Let 1≤ s< t be integers and let A_1, A_2, …, A_c be subsets with size s of [t], where c= t s. For given graphs G_1, G_2, …, G_c, the s-chromatic Ramsey number r^s, t(G_1, G_2, …, G_c), is the minimum positive integer N such that every t-coloring of E(K_N) yields a copy of G_i whose edges are colored by colors in the color set A_i for some i∈ [c]. The star-critical s-chromatic Ramsey number r_*^s, t(G_1, G_2, …, G_c), is the minimum integer ℓ such that every t-coloring of the edges in K_N- E(K_1, N- 1- ℓ) yields a copy of G_i whose edges are colored by colors in the color set A_i for some i∈ [c], where N= r^s, t(G_1, G_2, …, G_c). If G_1= G_2= …= G_c= G, then we simplify them to r^s, t(G) (also called the weakened Ramsey number) and r^s, t_*(G), respectively. In this paper, we determine all the values of r^s, t(K_1, m) and r_*^s, t(K_1, m), and part of the value of r^s, t(K_1, m_1, K_1, m_2, …, K_1, m_c).
We develop a circular-street argument, in the style of Pollak, to obtain a new proof that there are C_n = 1/n+12nn weakly increasing parking functions of length n ≥ 1, where C_n is the nth Catalan number.
For a sequence of tasks, each with a positive integer period, the pinwheel scheduling problem involves finding a valid schedule in the sense that the schedule performs one task per day and each task is performed at least once every consecutive days of its period. It had been conjectured by Chan and Chin in 1993 that there exists a valid schedule for any sequence of tasks with density, the sum of the reciprocals of each period, at most 5/6. Recently, Kawamura settled this conjecture affirmatively. In this paper we consider an extended version with real periods proposed by Kawamura, in which a valid schedule must perform each task i having a real period a_i at least l times in any consecutive ⌈ l a_i⌉ days for all positive integer l. We show that any sequence of tasks such that the periods take three distinct real values and the density is at most 5/6 admits a valid schedule. We hereby conjecture that the conjecture of Chan and Chin is true also for real periods.
We consider partially ordered sets of combinatorial structures under consecutive orders, meaning that two structures are related when one embeds in the other such that `consecutive' elements remain consecutive in the image. Given such a partially ordered set, we may ask decidability questions about its avoidance sets: subsets defined by a finite number of forbidden substructures. Two such questions ask, given a finite set of structures, whether its avoidance set is well quasi-ordered (i.e. contains no infinite antichains) or atomic (i.e. cannot be expressed as the union of two proper subsets). Extending some recent new approaches, we will establish a general framework, which enables us to answer these problems for a wide class of combinatorial structures, including graphs, digraphs and collections of relations.
In 2017, Clark Kimberling defined an interesting sequence B = 0100101100 & centerdot;& centerdot;& centerdot; of 0's and 1's by certain inflation rules, and he made a number of conjectures about this sequence and some related ones. In this note we prove his conjectures using, in part, the Walnut theorem-prover. We show how his word is related to the infinite Tribonacci word, and we determine both the factor complexity and critical exponent of B.
Following a recent paper of Anselmo et al., we consider m & times; n rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved with a finite automaton. We also generalize the result to every Sturmian characteristic word corresponding to a quadratic irrational. Finally, we also examine the analogous question for the Tribonacci word and the Thue-Morse word.
Locating-dominating codes have been studied widely since their introduction in the 1980s by Slater and Rall. In this paper, we concentrate on vertices that must belong to all minimum locating-dominating codes in a graph. We call them \emph{min-forced vertices}. We show that the number of min-forced vertices in a connected nontrivial graph of order $n$ is bounded above by $\frac{2}{3}\left(n -γ^{LD}(G)\right)$, where $γ^{LD}(G)$ denotes the cardinality of a minimum locating-dominating code. This implies that the maximum ratio between the number of min-forced vertices and the order of a connected nontrivial graph is at most $\frac{2}{5}$. Moreover, both of these bounds can be attained. In particular, the ratio $\frac{2}{5}$ is obtained by paths of order $5m$ having a unique minimum locating-dominating code of size $2m$. Furthermore, as a natural extension, we determine the number of different minimum locating-dominating codes in paths of all orders. In addition, we show that deciding whether a vertex is min-forced is co-NP-hard. 22 pages, 6 figures
It is known that any two trees on the same n leaves can be displayed by a network with n - 2 reticulations, and there are two trees that cannot be displayed by a network with fewer reticulations. But how many reticulations are needed to display multiple trees? For any set oft trees on n leaves, there is a trivial network with (t - 1)n reticulations that displays them. To do better, we have to exploit common structure of the trees to embed non-trivial subtrees of different trees into the same part of the network. In this paper, we show that, for t is an element of o(root log n), there is a set oft trees with virtually no common structure that could be exploited. More precisely, we show that, for any t is an element of o(root log n), there are t trees such that any network displaying them has (t - 1)n - o(n) reticulations. For t is an element of o(log n), we obtain a slightly weaker bound. We also prove that already for t = c log n, for any constant c > 0, there is a set of t trees that cannot be displayed by a network with o(n log n) reticulations, matching up to constant factors the known upper bound of O (n log n) reticulations sufficient to display all trees with n leaves. These results are based on simple counting arguments and extend to unrooted networks and trees.
In a recent paper, Francis, Illickan, Jose and Rajendraprasad showed that every n-vertex plane graph G has (under some natural restrictions) a vertex-partition into two sets V_1 and V_2 such that each V_i is dominating (every vertex of G contains a vertex of V_i in its closed neighbourhood) and face-hitting (every face of G is incident to a vertex of V_i). Their proof works by considering a supergraph G' of G that has certain properties, and among all such graphs, taking one that has the fewest edges. As such, their proof is not algorithmic. Their proof also relies on the 4-color theorem, for which a quadratic-time algorithm exists, but it would not be easy to implement. In this paper, we give a new proof that every n-vertex plane graph G has (under the same restrictions) a vertex-partition into two dominating face-hitting sets. Our proof is constructive, and requires nothing more complicated than splitting a graph into 2-connected components, finding an ear decomposition, and computing a perfect matching in a 3-regular plane graph. For all these problems, linear-time algorithms are known and so we can find the vertex-partition in linear time.
