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.
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 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.
Schu & uml;tzenberger's promotion operator a is a fundamental map in dynamical algebraic combinatorics. At first, its action was mainly considered on standard Young tableaux. But a was subsequently shown to have interesting properties when applied to natural labelings of other posets. Pechenik defined a K-theoretic version of promotion, aK, on mpacked labelings of tableaux. The operator aK was then extended to increasing labelings of other posets. The purpose of the current work is to show that the original action of aK on m-packed labelings yields interesting results when applied to partially ordered sets in general, and to rooted trees in particular. We show that under certain conditions, the sizes of the orbits and order of aK exhibit nice divisibility properties. We also completely determine, for certain values of m, the orbit sizes for the action on various types of rooted trees such as extended stars, combs, zippers, and a type of three-leaved tree.
In 1978, Anderson and White asked whether there is a decomposition of K12 into two graphs, one planar and one toroidal. Using theoretical arguments and a computer search of all maximal planar graphs of order 12, we show that no such decomposition exists. We further show that if G is planar of order 12 and H subset of G is toroidal, then H has at least two fewer edges than G. A computer search found all 123 unique pairs (G, H) that make this an equality.
We study infinite ternary words that contain few distinct palindromes. In particular, we classify such words according to their critical exponent.
A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It is known that the order $n$ and degree $d$ of a vertex-transitive nut graph satisfy $4 \mid d$, $d \ge 4$, $2 \mid n$ and $n \ge d + 4$; or $d \equiv 2 \pmod 4$, $d \ge 6$, $4 \mid n$ and $n \ge d + 6$. Here, we prove that for each such $n$ and $d$, there exists a $d$-regular Cayley nut graph of order $n$. As a direct consequence, we obtain all the pairs $(n, d)$ for which there is a $d$-regular vertex-transitive (resp. Cayley) nut graph of order $n$.
The symmetric random walk is known to be recurrent in one and two dimensions, and becomes transient in three or higher dimensions. We compare the symmetric random walk to walks driven by certain \polya\ urns. We show that, in contrast, if the probabilities of the random walk are instead driven by a \polya-Eggenberger urn, the states are recurrent only in one dimension. Further consideration of exchangeability reveals that the walk is null recurrent. As soon as the underlying Markov chain of \polya\ walk gets in two dimensions or higher, there is a positive probability that the walker gets lost in the space, and the probability of her recurrence is less than 1. On the other hand, a walk driven by Friedman urn behaves like the symmetric random walk, being recurrent in one and two dimensions and transient in higher dimensions. As Friedman urn scheme is not exchangeable, it is considerably harder to determine the nature of the recurrence in one and two dimensions. Empirical evidence through simulation suggests that in one dimension Friedman walk is positive recurrent.
Chip-firing is a combinatorial game on a graph, in which chips are placed and dispersed among its vertices until a stable configuration is achieved. We specifically study a chip-firing variant on an infinite, rooted, directed k-ary tree where we place kn chips labeled 0, 1, ... , kn 1 on the root for some nonnegative integer n, and we say a vertex v can fire if it has at least k chips. When a vertex fires, we select k labeled chips and send the ith smallest chip among them to its ith leftmost child. A stable configuration is reached when no vertex can fire. In this paper, we focus on stable configurations resulting from specific firing strategies based on permutations of 1, 2, ... , n. We then express the stable configuration as a permutation of 0, 1, 2, ... , kn 1 and explore its properties, such as the number of inversions and descents.
A new model for domination reconfiguration is introduced which combines the properties of the preexisting token addition/removal (TAR) and token sliding (TS) models. The vertices of the TARS-graph correspond to the dominating sets of $G$, where two vertices are adjacent if and only if they are adjacent via either the TAR reconfiguration rule or the TS reconfiguration rule. While the domination reconfiguration graph obtained by using only the TAR rule (sometimes called the dominating graph) will never have a Hamilton cycle, we show that for some classes of graphs $G$, by adding a relatively small number of token sliding edges, the resulting graph is not only hamiltonian, but is in fact pancyclic. In particular, if the underlying graphs are trees, complete graphs, or complete multipartite graphs, we show that their TARS-graphs will be pancyclic. Notably, we prove that if the TARS-graphs of $G$ and $H$ are pancyclic, then the TARS-graph of the join $G \vee H$ will also be pancyclic. We conclude by posing the question: Are all TARS-graphs pancyclic? 15 pages, 4 figures
In an oriented graph (G) over right arrow, the inversion of a subset X of vertices is the operation that reverses the orientation of all arcs with both end-vertices in X. The inversion graph of a graph G, denoted by I(G), is the graph whose vertices are orientations of G in which two orientations (G) over right arrow(1) and (G) over right arrow(2) are adjacent if and only if there is an inversion transforming (G) over right arrow(1) into (G) over right arrow(2). The inversion diameter of a graph G is the diameter of its inversion graph I(G), denoted by diam(I(G)). Havet, Horsch, and Rambaud (2024) first proved that for G of treewidth k, diam(Z(G)) <= 2k, and that there are graphs of treewidth k with inversion diameter k + 2 In this paper, we construct graphs of treewidth k with inversion diameter 2k, which implies that the previous upper bound diam(I(G)) <= 2k is tight. Moreover, for graphs with maximum degree Delta, Havet, Horsch, and Rambaud (2024) proved diam(I(G)) <= 2 Delta - 1 and conjectured that diam(I(G)) <= Delta. We prove the conjecture when Delta = 3 with the help of computer calculations.
We prove limit theorems for the number of fixed points occurring in a random pattern-avoiding permutation distributed according to a one-parameter family of biased distributions. The bias parameter exponentially tilts the distribution towards favoring permutations with more or fewer fixed points than is typical under the uniform distribution. One case we study features a phase transition where the limiting distribution changes abruptly from negative binomial to Rayleigh to normal depending on the bias parameter.
The problem of realizing a given degree sequence by a multigraph can be thought of as a relaxation of the classical degree realization problem (where the realizing graph is simple). This paper concerns the case where the realizing multigraph is required to be bipartite. The problem of characterizing degree sequences that can be realized by a bipartite (simple) graph has two variants. In the simpler one, termed BDR $$^P$$ , the partition of the degree sequence into two sides is given as part of the input. A complete characterization for realizability in this variant was given by Gale and Ryser over sixty years ago. However, the variant where the partition is not given, termed BDR, is still open. For bipartite multigraph realizations, there are again two variants. For BDR $$^P$$ , where the partition is given as part of the input, a complete characterization was known for determining whether the bi-sequence is r-max-bigraphic, namely, if there is a multigraph realization whose underlying graph is bipartite, such that the maximum number of copies of an edge is at most r. We present a complete characterization for determining if there is a bipartite multigraph realization such that the total number of excess edges is at most t. As for the variant BDR, where the partition is not given, we show that determining whether a given (single) sequence admits a bipartite multigraph realization is NP-hard. On the positive side, we provide an algorithm that computes optimal realizations for the case where the number of balanced partitions is polynomial, and present sufficient conditions for the existence of bipartite multigraph realizations that depend only on the largest degree of the sequence.
In the Fully Leafed Induced Subtrees, one is given a graph G and two integers a and b and the question is to find an induced subtree of G with a vertices and at least b leaves. This problem is known to be NP-complete even when the input graph is 4-regular. Polynomial algorithms are known when the input graph is restricted to be a tree or series-parallel. In this paper we generalize these results by providing an FPT algorithm parameterized by treewidth. We also provide a polynomial algorithm when the input graph is restricted to be a chordal graph.
