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 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.
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.
