Search results for "Computer Science - Discrete Mathematics"
showing 7 items of 47 documents
Minimal forbidden factors of circular words
2017
Minimal forbidden factors are a useful tool for investigating properties of words and languages. Two factorial languages are distinct if and only if they have different (antifactorial) sets of minimal forbidden factors. There exist algorithms for computing the minimal forbidden factors of a word, as well as of a regular factorial language. Conversely, Crochemore et al. [IPL, 1998] gave an algorithm that, given the trie recognizing a finite antifactorial language $M$, computes a DFA recognizing the language whose set of minimal forbidden factors is $M$. In the same paper, they showed that the obtained DFA is minimal if the input trie recognizes the minimal forbidden factors of a single word.…
Completely independent spanning trees in some regular graphs
2014
International audience; Let k >= 2 be an integer and T-1,..., T-k be spanning trees of a graph G. If for any pair of vertices {u, v} of V(G), the paths between u and v in every T-i, 1 <= i <= k, do not contain common edges and common vertices, except the vertices u and v, then T1,... Tk are completely independent spanning trees in G. For 2k-regular graphs which are 2k-connected, such as the Cartesian product of a complete graph of order 2k-1 and a cycle, and some Cartesian products of three cycles (for k = 3), the maximum number of completely independent spanning trees contained in these graphs is determined and it turns out that this maximum is not always k. (C) 2016 Elsevier B.V. All righ…
Almost disjoint spanning trees: relaxing the conditions for completely independent spanning trees
2017
International audience; The search of spanning trees with interesting disjunction properties has led to the introduction of edge-disjoint spanning trees, independent spanning trees and more recently completely independent spanning trees. We group together these notions by dening (i, j)-disjoint spanning trees, where i (j, respectively) is the number of vertices (edges, respectively) that are shared by more than one tree. We illustrate how (i, j)-disjoint spanning trees provide some nuances between the existence of disjoint connected dominating sets and completely independent spanning trees. We prove that determining if there exist two (i, j)-disjoint spanning trees in a graph G is NP-comple…
Grand Dyck paths with air pockets
2022
Grand Dyck paths with air pockets (GDAP) are a generalization of Dyck paths with air pockets by allowing them to go below the $x$-axis. We present enumerative results on GDAP (or their prefixes) subject to various restrictions such as maximal/minimal height, ordinate of the last point and particular first return decomposition. In some special cases we give bijections with other known combinatorial classes.
Topological properties of cellular automata on trees
2012
We prove that there do not exist positively expansive cellular automata defined on the full k-ary tree shift (for k>=2). Moreover, we investigate some topological properties of these automata and their relationships, namely permutivity, surjectivity, preinjectivity, right-closingness and openness.
On List Coloring with Separation of the Complete Graph and Set System Intersections
2022
We consider the following list coloring with separation problem: Given a graph $G$ and integers $a,b$, find the largest integer $c$ such that for any list assignment $L$ of $G$ with $|L(v)|= a$ for any vertex $v$ and $|L(u)\cap L(v)|\le c$ for any edge $uv$ of $G$, there exists an assignment $\varphi$ of sets of integers to the vertices of $G$ such that $\varphi(u)\subset L(u)$ and $|\varphi(v)|=b$ for any vertex $u$ and $\varphi(u)\cap \varphi(v)=\emptyset$ for any edge $uv$. Such a value of $c$ is called the separation number of $(G,a,b)$. Using a special partition of a set of lists for which we obtain an improved version of Poincar\'e's crible, we determine the separation number of the c…
Asymptotic bit frequency in Fibonacci words
2021
It is known that binary words containing no $k$ consecutive 1s are enumerated by $k$-step Fibonacci numbers. In this note we discuss the expected value of a random bit in a random word of length $n$ having this property.